1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
2.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space, the surface of a completely round ball. The given point is the center of the mathematical ball. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the ball. The sphere share the same radius, diameter, center. The area of a sphere is: A = 4 π r 2. The total volume is the summation of all shell volumes: V ≈ ∑ A ⋅ r. In the limit as δr approaches zero this equation becomes: V = ∫ 0 r A d r ′. Substitute V: 4 3 π r 3 = ∫ 0 r A d r ′. Differentiating both sides of this equation with respect to r yields A as a function of r: 4 π r 2 = A. Which is generally abbreviated as: A = 4 π r 2. Alternatively, the element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. For more generality, see element. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. The total volume is the summation of all incremental volumes: V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes: V = ∫ − r r π y 2 d x.
Sphere
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Circumscribed cylinder to a sphere
Sphere
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A two-dimensional perspective projection of a sphere
Sphere
Sphere
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Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
3.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, three-dimensional space. When working exclusively in Euclidean space, the definite article is used, so, the plane refers to the whole space. Fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane. Euclid set forth the great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined postulates which he then used to prove various geometrical statements. In his work Euclid never makes use of numbers to measure length, area. In this way the Euclidean plane is not quite the same as the Cartesian plane. This section is solely concerned with planes embedded in three dimensions: specifically, in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines. A line is either parallel to a plane, is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other.
Plane (geometry)
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Vector description of a plane
Plane (geometry)
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Two intersecting planes in three-dimensional space
4.
History of geometry
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Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced the axiomatic method still in use today. A modern mathematician might be hard put to derive some of them without the use of calculus. This assumes that π is × ², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an octagon with an area of 63 units. This gave a second value for π of 3.111... The two problems together indicate a range of values for π between 3.11 and 3.16. The Babylonians may have known the general rules for measuring volumes. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 1/8. The Babylonians are also known for the Babylonian mile, today.
History of geometry
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Part of the " Tab.Geometry. " (Table of Geometry) from the 1728 Cyclopaedia.
History of geometry
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Rigveda manuscript in Devanagari.
History of geometry
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Statue of Euclid in the Oxford University Museum of Natural History.
History of geometry
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Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310)
5.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in deducing other propositions from these. The Elements begins with geometry, still taught as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates. The Elements is mainly a systematization of earlier knowledge of geometry. There are 13 total books in the Elements: I -- IV and VI discuss geometry. Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as irrational numbers are introduced. The infinitude of prime numbers is proved.
Euclidean geometry
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Detail from Raphael 's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction.
Euclidean geometry
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A surveyor uses a level
Euclidean geometry
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Sphere packing applies to a stack of oranges.
Euclidean geometry
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Geometry is used in art and architecture.
6.
Spherical geometry
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Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry, not Euclidean. Two practical applications of the principles of spherical geometry are astronomy. In geometry, the basic concepts are points and lines. On a sphere, points are defined in the usual sense. On a sphere, the geodesics are the great circles; other geometric concepts are defined as with straight lines replaced by great circles. Shares with that geometry the property that a line has no parallels through a given point. An important geometry related to that of the sphere is that of the real plane; it is obtained by identifying antipodal points on the sphere. It has different global properties. In particular, it one-sided. Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas. Spherical geometries exist; see elliptic geometry. The book of unknown arcs of a sphere written by the Islamic Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the solution of a spherical triangle by means of the polar triangle. The book by Regiomontanus, written around 1463, is the first pure trigonometrical work in Europe.
Spherical geometry
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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.
7.
Non-Euclidean geometry
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. The essential difference between the metric geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ. In elliptic geometry the lines "curve toward" each other and intersect. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4.
Non-Euclidean geometry
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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometry
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Projecting a sphere to a plane.
8.
Elliptic geometry
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Elliptic geometry has a variety of properties that differ from those of classical Euclidean geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at the absolute pole of the given line. There are no antipodal points in elliptic geometry. Every point corresponds to an polar line of which it is the absolute pole. Any point on this polar line forms an absolute pair with the pole. The distance between them is a quadrant. The distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called a hyperbola according as it has no asymptote or two asymptotes. A simple way to picture elliptic geometry is to look at a globe. They intersect at the poles. With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line.
Elliptic geometry
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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.
9.
Hyperbolic geometry
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In mathematics, hyperbolic geometry is a non-Euclidean geometry. A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and space. In Russia it is commonly called Lobachevskian geometry after one of the Russian geometer Nikolai Lobachevsky. This page is mainly about the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases. Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, hyperbolic. All theorems including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Euclid's Elements prove the existence of parallel/non-intersecting lines. This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have exactly the same properties as straight lines in Euclidean geometry. For example, lines can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry.
Hyperbolic geometry
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A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring
Hyperbolic geometry
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Lines through a given point P and asymptotic to line R
Hyperbolic geometry
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A coral with similar geometry on the Great Barrier Reef
Hyperbolic geometry
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M.C. Escher 's Circle Limit III, 1959
10.
Synthetic geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the tools directly related to them, compass and straightedge, to draw conclusions and solve problems. Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach from other approaches. Other approaches to geometry are embodied in algebraic geometries, where one would use analysis and algebraic techniques to obtain geometric results. Geometry, as presented in the elements, is the quintessential example of the use of the synthetic method. It was the favoured method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective non-Euclidean geometries. For example the geometer Jakob Steiner always gave preference to synthetic methods. The process of logical synthesis begins with some definite starting point. This starting point is primitives and axioms about these primitives: Primitives are the most basic ideas. Typically they include both relationships. The terms themselves are undefined. Axioms are statements about these primitives; for example, any two points are together incident with just one line. Axioms are assumed true, not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
Synthetic geometry
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Projecting a sphere to a plane.
11.
Analytic geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian system is applied to manipulate equations for planes, straight lines, squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean Euclidean space. The numerical output, however, might also be a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations. Coordinates, equations were subsidiary notions applied to a specific geometric situation. Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. The alternative term used for analytic geometry, is named after Descartes. This work, written in its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartes's masterpiece receive due recognition.
Analytic geometry
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Cartesian coordinates
12.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the points at infinity. More advanced questions involve relations between the curves given by different equations. Algebraic geometry has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas. The study of the real points of an algebraic variety is the subject of algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties. With the rise of the computers, a algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for finding the properties of explicitly given algebraic varieties. This means that a point of such a scheme may be either a subvariety. This approach also enables a unification of classical algebraic geometry mainly concerned with complex points, of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.
Algebraic geometry
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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
13.
Riemannian geometry
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This gives, in local notions of angle, length of curves, volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. It is a very abstract generalization of the geometry of surfaces in R3. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary including the standard types of Non-Euclidean geometry. Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Disclinations produce curvature. The choice is made depending on simplicity of formulation. Most of the results can be found by D. Ebin. The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.
Riemannian geometry
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Bernhard Riemann
Riemannian geometry
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Projecting a sphere to a plane.
14.
Differential geometry
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Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. Differential geometry developed to the mathematical analysis of surfaces. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address. Initially applied to the Euclidean space, further explorations led to non-Euclidean space, metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a symmetric bilinear form defined on the tangent space at each point. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
15.
Symplectic geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds;, differentiable manifolds equipped with a closed, nondegenerate 2-form. A symplectic geometry is defined on a even-dimensional space, a differentiable manifold. On this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures angles, the symplectic form measures areas. An example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the momentum p, which form a point in the Euclidean plane ℝ2. The area is important because as dynamical systems evolve in time, this area is invariant. Higher dimensional symplectic geometries are defined analogously. Symplectic geometry has a number of differences from Riemannian geometry, the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors. Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are topological restrictions. For example, every symplectic manifold is orientable. Both concepts play a fundamental role in their respective disciplines. Every Kähler manifold is also a symplectic manifold.
Symplectic geometry
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Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.
16.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains many points. A geometry based on the graphics displayed on a screen, where the pixels are considered to be the points, would be a finite geometry. Finite geometries can also be defined purely axiomatically. However, dimension two has projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes. There are two main kinds of finite geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Finite projective plane geometry may be described by fairly simple axioms. There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry. The simplest plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points so this plane contains six lines.
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
17.
Projective geometry
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Projective geometry is a topic of mathematics. Projective is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, a selective set of basic geometric concepts. The first issue for geometers is what kind of geometry is adequate for a situation. One source for projective geometry was indeed the theory of perspective. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex space, the coordinates used being complex numbers. Major types of more abstract mathematics were based on projective geometry. Projective was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions Projective begins with the study of configurations of lines.
Projective geometry
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Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
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Projecting a sphere to a plane.
Projective geometry
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Forms
18.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. The inside of a cube, a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one, found necessary to describe electromagnetism. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with gravity. The state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in the sciences. In mathematics, the dimension of an object is an intrinsic independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question "what makes En n-dimensional?" For example, this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces. A tesseract is an example of a four-dimensional object.
Dimension
19.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every constructible using straightedge and compass may also be constructed using compass alone. A number of ancient problems in plane geometry impose this restriction. In some cases were unable to do so. Gauss showed that most are not. Some of the most famous problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: a point on the circle. The compass may not collapse when it's not drawing a circle. It has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to extend an existing segment. Several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass-and-straightedge construction
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A compass
Compass-and-straightedge construction
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Creating a regular hexagon with a ruler and compass
20.
Angle
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This plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined at the point of intersection. Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. The angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος, meaning "crooked, curved," and the English word "ankle". Both are connected with * ank -, meaning "to bend" or "bow". According to Proclus an angle must be a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted B A C ^.
Angle
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An angle enclosed by rays emanating from a vertex.
21.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero. Various disciplines within mathematics have given different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space, locally homeomorphic to a line. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in mathematical fields. Closely related meanings include the graph of a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study. This can be seen on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, by a stick in the sand on a beach. Historically, the term "line" was used in place of the more modern term "curve". Hence "right line" were used to distinguish what are today called lines from "curved lines". Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points,". Later commentators further classified lines according to various schemes.
Curve
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Megalithic art from Newgrange showing an early interest in curves
Curve
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A parabola, a simple example of a curve
22.
Diagonal
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In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. In algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also non-mathematical uses. As applied to a polygon, a diagonal is a segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For re-entrant polygons, some diagonals are outside of the polygon. For n-gons with n=3. 4... the number of regions is 1, 4, 11, 25, 50, 91, 154, 246... This is OEIS sequence A006522. The off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero. A superdiagonal entry is one, directly to the right of the main diagonal. Just as diagonal entries are those A i j with j = i, the superdiagonal entries are those with 1. In geometric studies, the idea of intersecting the diagonal with itself is not directly, but by perturbing it within an equivalence class.
Diagonal
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A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure
Diagonal
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The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length, while AC (shown in red) is a face diagonal and has length.
23.
Orthogonal
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, engineering. The word comes from the Greek ὀρθός, meaning "upright", γωνία, meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y. The largest subspace of V, orthogonal to a given subspace is its orthogonal complement. Two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
Orthogonal
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The line segments AB and CD are orthogonal to each other.
24.
Perpendicular
–
In elementary geometry, the property of being perpendicular is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects. A line is said to be perpendicular to another line if the two lines intersect at a right angle. For this reason, we may speak as being perpendicular without specifying an order. Perpendicularity easily extends to rays. In symbols, A B ¯ ⊥ C D ¯ means segment AB is perpendicular to line segment CD. A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines. Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle. Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of geometric objects. The word "foot" is frequently used with perpendiculars. This usage is exemplified in the top diagram, above, its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom. Step 2: construct circles centered at A' and B' having equal radius.
Perpendicular
–
The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
25.
Parallel (geometry)
–
By extension, a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the three-dimensional space that never meet. Parallel lines are the subject of Euclid's parallel postulate. Euclidean space is a special instance of this type of geometry. Some other spaces, such as hyperbolic space, have analogous properties that are sometimes referred to as parallelism. The parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to CD. In the Unicode set, the "parallel" and "not parallel" signs have codepoints U +2225 and U +2226, respectively. In addition, U +22 D5 represents the relation "parallel to". Line m is in the same plane as line l but does not intersect l. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are then consequences of Euclid's Parallel Postulate. Another property that also involves measurement is that lines parallel to each other have the same gradient. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate.
Parallel (geometry)
–
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.
26.
Vertex (geometry)
–
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. However, in theory, vertices may have fewer than two incident edges, usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices, at the points where its curvature is minimal. There are two types of principal vertices: mouths. A principal xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P. According to the two ears theorem, every simple polygon has at least two ears. A principal xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. This equation is known as Euler's formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 6 faces, hence 8 vertices. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex".
Vertex (geometry)
–
A vertex of an angle is the endpoint where two line segments or rays come together.
27.
Congruence (geometry)
–
This means that either object can be reflected so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. In elementary geometry the congruent is often used as follows. The word equal is often used in place of congruent for these objects. Two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure. Two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size but not in shape. For two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have numerically identical sequences side-angle-side-angle... for n sides and n angles. Congruence of polygons can be established graphically as follows: First, label the corresponding vertices of the two figures. Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.
Congruence (geometry)
–
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.)
28.
Similarity (geometry)
–
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained with additional translation, rotation and reflection. This means that either object can be rescaled, reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular scaling of the other. For example, all circles are similar to each other, all equilateral triangles are similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding angles of similar polygons have the same measure. It can be shown that two triangles having congruent angles are similar, the corresponding sides can be proved to be proportional. This is known as the AAA theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of, sufficient for two triangles to be similar: 1. The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. 2. All the corresponding sides have lengths in the same ratio: AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other.
Similarity (geometry)
–
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)
–
Figures shown in the same color are similar
29.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry if it can be translated without changing its overall shape. An object rotated along a line known as a screw axis. An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry. A dyadic R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Or, biconditional, nand, nor.
Symmetry
–
Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
Symmetry
Symmetry
–
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Symmetry
–
The ceiling of Lotfollah mosque, Isfahan, Iran has 8-fold symmetries.
30.
One-dimensional space
–
In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the line, where the position of each point on it can be described by a single number. The real projective line are one-dimensional, though the latter is homeomorphic to a circle. In algebraic geometry there are several structures which referred to in other terms. For a k, it is a one-dimensional vector space over itself. Similarly, the projective line over k is a one-dimensional space. More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. The only regular polytope in one dimension is the segment, with the Schläfli symbol. The hypersphere in 1 dimension is a pair of points, sometimes called a 0-sphere as its surface is zero-dimensional. Its length is L = 2 r where r is the radius. The most popular coordinate systems are the angle.
One-dimensional space
–
Number line
31.
Point (geometry)
–
In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". Further generalizations are represented by an ordered tuplet of n terms, where n is the dimension of the space in which the point is located. Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. Similar constructions exist that define the plane, other related concepts. By the way, a degenerate segment consists of only one point. In spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics. In all of the common definitions, a point is 0-dimensional.
Point (geometry)
–
Projecting a sphere to a plane.
32.
Line (geometry)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. The straight line is that, equally extended between its points." Given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is usually left undefined. The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also, for example, permits physicists to think of the path of a light ray as being a line. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive. In those situations where a line is a defined concept, as in coordinate geometry, some fundamental ideas are taken as primitives. When the concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. Descriptions of this type may be referred to, as definitions in this informal style of presentation.
Line (geometry)
–
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).
33.
Line segment
–
A closed segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a square. When the end points both lie on a curve such as a circle, a segment is called a chord. Sometimes one needs to distinguish between "open" and "closed" line segments. Equivalently, a segment is the convex hull of two points. Thus, the segment can be expressed as a convex combination of the segment's two end points. Thus in R 2 the segment with endpoints A = and C = is the following collection of points:. A segment is a connected, non-empty set. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, parallel, none of these. Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a segment. As a degenerate orbit this is a elliptic trajectory. In addition to appearing as the diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Line segment
–
historical image – create a line segment (1699)
34.
Ray (geometry)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. The straight line is that, equally extended between its points." Given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is usually left undefined. The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also, for example, permits physicists to think of the path of a light ray as being a line. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive. In those situations where a line is a defined concept, as in coordinate geometry, some fundamental ideas are taken as primitives. When the concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. Descriptions of this type may be referred to, as definitions in this informal style of presentation.
Ray (geometry)
–
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).
35.
Length
–
In geometric measurements, length is the most extended dimension of an object. In the International System of Quantities, length is any quantity with distance. In other contexts "length" is the measured dimension of an object. For example, it is possible to cut a length of a wire, shorter than thickness. Volume is a measure of three dimensions. In most systems of measurement, the unit of length is a unit, from which other units are defined. Measurement has been important ever since humans started using building materials, occupying land and trading with neighbours. As society has become more technologically oriented, much higher accuracies of measurement are required from micro-electronics to interplanetary ranging. This added together to make longer units like the stride. The cubit could vary considerably due to the different sizes of people. After Albert Einstein's special relativity, length can longer be thought of being constant in all reference frames. This means length of an object is variable depending on the observer. In the physical sciences and engineering, when one speaks of "units of length", the word "length" is synonymous with "distance". There are several units that are used to measure length. In the International System of Units, the basic unit of length is now defined in terms of the speed of light.
Length
–
Base quantity
36.
Two-dimensional space
–
In physics and mathematics, two-dimensional space or bi-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called width. Both directions lie in the same plane. A sequence of real numbers can be understood as a location in n-dimensional space. When n = 2, the set of all such locations is called bi-dimensional space, usually is thought of as a Euclidean space. Both authors have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Later, the plane was thought as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by mathematician Caspar Wessel. Argand diagrams are frequently used to plot the positions of the zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in two-dimensional space by means of two coordinates. Two coordinate axes are given which cross each other at the origin. They are usually labeled x and y.
Two-dimensional space
–
Bi-dimensional Cartesian coordinate system
37.
Area
–
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. It is the two-dimensional analog of the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the area of any other shape or surface is a dimensionless real number. There are well-known formulas for the areas of simple shapes such as triangles, rectangles, circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cylinder, the area of its boundary surface is called the surface area. Area plays an important role in modern mathematics. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. It can be proved that such a function exists.
Area
–
A square metre quadrat made of PVC pipe.
Area
–
The combined area of these three shapes is approximately 15.57 squares.
38.
Polygon
–
The points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The geometrical notion of a polygon has been adapted in various ways to suit particular purposes. They often define a polygon accordingly. A polygonal boundary may be allowed creating star polygons and other self-intersecting polygons. Other generalizations of polygons are described below. The word "polygon" derives from the Greek adjective πολύς "many" and γωνία "corner" or "angle". It has been suggested that γόνυ "knee" may be the origin of “gon”. Polygons are primarily classified by the number of sides. See table below. Polygons may be characterized by their type of non-convexity: Convex: any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any segment with endpoints on the boundary passes through only interior points between its endpoints.
Polygon
–
Historical image of polygons (1699)
Polygon
–
Some different types of polygon
Polygon
–
The Giant's Causeway, in Northern Ireland
39.
Triangle
–
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral isosceles triangles. The 45 -- 45 -- 90 right triangle, which appears in the square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles and geometric figures to identify sides of equal lengths. In a triangle, the pattern is usually no more than 3 ticks.
Triangle
–
The Flatiron Building in New York is shaped like a triangular prism
Triangle
–
A triangle
40.
Altitude (triangle)
–
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the altitude. The length of the altitude, simply called the altitude, is the distance between the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude of that vertex. It is a special case of orthogonal projection. Thus the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometric functions. In an isosceles triangle, the altitude having the incongruent side as its base will have the midpoint of that side as its foot. Also the altitude having the incongruent side as its base will form the angle bisector of the vertex. It is common to mark the altitude with the letter h, often subscripted with the name of the side the altitude comes from. In a right triangle, the altitude with the hypotenuse c as base divides the hypotenuse into two lengths p and q. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex of the right angle.
Altitude (triangle)
–
Three altitudes intersecting at the orthocenter
41.
Hypotenuse
–
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse is the square root of 25, 5. The ὑποτείνουσα was used by many other ancient authors. A folk etymology says that tenuse means "side", so hypotenuse means a support like a prop or buttress, but this is inaccurate. The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Many computer languages support the ISO C standard function hypot, which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate. Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. The angle returned will normally be that given by atan2. Orthographic projections: The length of the hypotenuse equals the sum of the lengths of the orthographic projections of both catheti. Cathetus Triangle Space diagonal Nonhypotenuse number Taxicab geometry Trigonometry Special right triangles Pythagoras Hypotenuse at Encyclopaedia of Mathematics Weisstein, Eric W. "Hypotenuse". MathWorld.
Hypotenuse
–
A right-angled triangle and its hypotenuse.
42.
Parallelogram
–
In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology reflects the definition. Square – A parallelogram with four sides of equal length and angles of equal size. Two pairs of opposite angles are equal in measure. The diagonals bisect each other. One pair of opposite sides are parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles. The sum of the squares of the sides equals the sum of the squares of the diagonals. It has rotational symmetry of order 2. The sum of the distances from any interior point to the sides is independent of the location of the point. Opposite sides of a parallelogram are parallel and so will never intersect.
Parallelogram
–
This parallelogram is a rhomboid as it has no right angles and unequal sides.
43.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ ABCD. Opposite sides of a square are both equal in length. All four angles of a square are equal. All four sides of a square are equal. The diagonals of a square are equal. The square is the n = 2 case of the families of n-orthoplexes. A square has Schläfli symbol. T, is an octagon. H, is a digon. The area A is A = ℓ 2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 2.
Square
–
A regular quadrilateral (tetragon)
44.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD. The rectangle comes from the Latin rectangulus, a combination of rectus and angulus. A crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. Its angles are not right angles. Other geometries, such as spherical, hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling a rectangle by polygons. A convex quadrilateral with successive sides b, c, d whose area is 1 2. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium in which both pairs of opposite sides equal in length. A trapezium is a convex quadrilateral which has at least one pair of opposite sides.
Rectangle
–
Running bond
Rectangle
–
Rectangle
Rectangle
–
Basket weave
45.
Rhombus
–
In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Every rhombus is a kite. A rhombus with right angles is a square. The word "rhombus" comes from Greek ῥόμβος, meaning "to turn round and round". The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two circular cones sharing a common base. The surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. The vertices are at and. This is a special case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular;, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite angles. The first property implies that every rhombus is a parallelogram.
Rhombus
–
Some polyhedra with all rhombic faces
Rhombus
–
Two rhombi.
Rhombus
46.
Rhomboid
–
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of equal length is a rhombus but not a rhomboid. A parallelogram with angled corners is a rectangle but not a rhomboid. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in terminology referring to both its two - and three-dimensional meaning. And let quadrilaterals other than these be called trapezia. Heath suggests that rhomboid was an older term already in use. It has rotational symmetry of order 2. In biology, rhomboid may describe a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves or cephalopod fins. In a type of arthritis called pseudogout, crystals of calcium dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope. Weisstein, Eric W. "Rhomboid". MathWorld.
Rhomboid
–
These shapes are rhomboids
47.
Quadrilateral
–
In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle, sometimes tetragon for consistency with pentagon, hexagon and so on. The origin of the word "quadrilateral" is the two Latin words quadri, latus, meaning "side". Quadrilaterals are complex, also called crossed. Simple quadrilaterals are either concave. This is a special case of the n-gon interior angle formula × 180 °. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges. Any quadrilateral, not self-intersecting is a simple quadrilateral. In a quadrilateral, all interior angles are less than 180 ° and the two diagonals both lie inside the quadrilateral. Irregular quadrilateral or trapezium: no sides are parallel. Trapezium or trapezoid: at least one pair of opposite sides are parallel. Trapezoids include parallelograms. Isosceles trapezium or isosceles trapezoid: the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting a trapezoid with diagonals of equal length. Parallelogram: a quadrilateral with two pairs of parallel sides.
Quadrilateral
–
Six different types of quadrilaterals
48.
Trapezoid
–
The other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below. The recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclid's Elements. This article uses the term trapezoid in the sense, current in the United States and Canada. In other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles. Right trapezoids are used for estimating areas under a curve. The base angles have the same measure. It has symmetry. An obtuse trapezoid with two pairs of parallel sides is a parallelogram. A parallelogram has central rotational symmetry. A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle. There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids.
Trapezoid
–
The Temple of Dendur in the Metropolitan Museum of Art in New York City
Trapezoid
–
Trapezoid
Trapezoid
–
Example of a trapeziform pronotum outlined on a spurge bug
49.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, they are opposite to each other rather than adjacent. Kite quadrilaterals are named for flying kites, which often have this shape and which are in turn named for a bird. The word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. The word "kite" is often restricted to the convex variety. A concave kite is a type of pseudotriangle. If all four sides of a kite have the same length, it must be a rhombus. If a kite is equiangular, meaning that all four of its angles are equal, then it must also thus a square. A kite with three equal 108 ° angles and the convex hull of the lute of Pythagoras. The kites that are also cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two equal angles on opposite sides of the symmetry axis are each 90 degrees. They are in fact bicentric quadrilaterals. Among all the bicentric quadrilaterals with a given two radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the trihexagonal tiling. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with 12, 5π / 6, 5π / 12.
Kite (geometry)
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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
50.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the centre is called the radius. A circle is a closed curve which divides the plane into two regions: an interior and an exterior. The bounding line is called the point, its centre. Annulus: the ring-shaped object, the region bounded by two concentric circles. Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circumference: the length of one circuit along the circle, or the distance around the circle. It is twice the radius. Disc: the region of the plane bounded by a circle. Lens: the intersection of two discs. Passant: a coplanar straight line that does not touch the circle. Sector: a region bounded by two radii and an arc lying between the radii. Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints.
Circle
–
The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo
Circle
–
A circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (O) in magenta.
Circle
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Circular piece of silk with Mongol images
Circle
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Circles in an old Arabic astronomical drawing.
51.
Diameter
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It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. The word "diameter" is derived from Greek διάμετρος, "diameter of a circle", from δια -, "across, through" "measure". It is often abbreviated DIA, dia, ⌀. In more modern usage, the length of a diameter is also called the diameter. D = 2 r r = d 2. Both quantities can be calculated efficiently using rotating calipers. For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse. The longest diameter is called the major axis. The definitions given above are only valid for circles, convex shapes. So, if A is the subset, the diameter is sup. If the distance function d is viewed here as having R, this implies that the diameter of the empty set equals − ∞. In geometry, the diameter is an important global Riemannian invariant. The variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke.
Diameter
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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.
52.
Circumference
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The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of special importance in geometry and trigonometry. Informally "circumference" may also refer to the edge itself rather than to the length of the edge. The circumference of a circle is the distance around it. The term is used when measuring physical objects, well as when considering geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics. Pi, is represented by the Greek π. The numerical value of π is 3.14159 26535 89793.... The above formula can be rearranged to solve for the circumference: C = π ⋅ d = 2 r. The use of the constant π is ubiquitous in mathematics, science. The constant ratio of circumference to radius C r = 2 π also has many uses in mathematics, science. These uses are not limited to radians, physical constants. The Greek τ is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the complete elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph.
Circumference
–
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.
53.
Area of a circle
–
In geometry, the area enclosed by a circle of radius r is πr2. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using its more sophisticated offspring, real analysis. However the area of a disk was studied by the Ancient Greeks. Eudoxus of Cnidus in the fifth B.C. had found that the area of a disk is proportional to its radius squared. The area of a triangle is half the base times the height, yielding the area πr2 for the disk. A variety of arguments have been advanced historically to establish the A = π r 2 of varying degrees of mathematical rigor. The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius. If the area of the circle is not equal to that of the triangle, then it must be less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way. Suppose that the area C enclosed by the circle is greater than the area T 1⁄2 cr of the triangle.
Area of a circle
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Circle with square and octagon inscribed, showing area gap
54.
Three-dimensional space
–
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the dimension. In mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called Euclidean space. It is commonly represented by the ℝ3. This serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in this case, these three values can be labeled by any combination of three chosen from height, depth, breadth. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular at the origin, the point at which they cross. They are usually labeled x, y, z. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane.
Three-dimensional space
–
Three-dimensional Cartesian coordinate system with the x -axis pointing towards the observer. (See diagram description for correction.)
55.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the cubic metre. Three mathematical shapes are also assigned volumes. Circular shapes can be easily calculated using arithmetic formulas. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape's boundary. Two-dimensional shapes are assigned zero volume in the three-dimensional space. The volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the combined volume is not additive. In geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length. In the International System of Units, the standard unit of volume is the cubic metre.
Volume
–
A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
56.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0.5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-sparking qualities. Beryllium has excellent metalworking, machining properties. Beryllium has specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, aerospace. Beryllium alloys present a toxic hazard during manufacture. Beryllium copper is a ductile, machinable alloy. Beryllium is resistant to non-oxidizing acids, to plastic decomposition products, to galling. Beryllium can be heat-treated for increased strength, electrical conductivity. Beryllium copper attains the greatest strength of any copper-based alloy. In as finished objects, beryllium copper presents no known health hazard. However, inhalation of dust, fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the bloodstream. The International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen. Beryllium copper is a non-ferrous alloy used in springs, spring wire, other parts that must retain their shape under repeated stress and strain.
Cube
–
Example of a non-sparking tool made of beryllium copper
Cube
57.
Cuboid
–
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum. In a rectangular cuboid, opposite faces of a cuboid are equal. The terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms "rectangular prism" and "prism", however, are ambiguous, since they do not specify all angles. The square cuboid, right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, its symmetry is doubled from to, order 16. The cube is a special case of the square cuboid in which all six faces are squares. Its symmetry is raised from, to, order 48. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are often used for boxes, cupboards, buildings, etc.. Cuboids are among those solids that can tessellate 3-dimensional space. A cuboid with integer edges well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.
Cuboid
–
Rectangular cuboid
58.
Cylinder (geometry)
–
It is one of the most basic curvilinear geometric shapes. If the ends are open, it is called an open cylinder. If the ends are closed by flat surfaces it is called a solid cylinder. The volume of such a cylinder have been known since deep antiquity. The area of the side is also known as L. An open cylinder therefore has surface area L = 2πrh. The area of a closed cylinder is made up the sum of all three components: top, bottom and side. Its area is A = 2πr2 + 2πrh = 2πr = πd = L +2 B, where d is the diameter. For a given volume, the closed cylinder with the smallest area has h = 2r. Equivalently, for a given area, the closed cylinder with the largest volume has h = 2r, i.e. the cylinder fits snugly in a cube. Cylindric sections are the intersections of cylinders with planes. For a circular cylinder, there are four possibilities. A tangent to the cylinder meets the cylinder in a single straight line segment. Moved to itself, the plane either does not intersect the cylinder or intersects it in two parallel line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, in a circle.
Cylinder (geometry)
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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)
–
In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
59.
Pyramid (geometry)
–
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base apex form a triangle, called a lateral face. It is a solid with polygonal base. A pyramid with an n-sided base will have n + 1 vertices, 2n edges. All pyramids are self-dual. A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid, like the physical pyramid structures. A triangle-based pyramid is more often called a tetrahedron. A right-angled pyramid has its apex above an vertex of the base. In a tetrahedron these qualifiers will change based on which face is considered the base. Pyramids are a subclass of the prismatoids. Pyramids can be doubled by adding a second offset point on the other side of the base plane. A right pyramid with a regular base has isosceles triangle sides, with symmetry is Cnv or, with order 2n.
Pyramid (geometry)
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Regular-based right pyramids
60.
Four-dimensional space
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In mathematics, four-dimensional space is a geometric space with four dimensions. It typically is more specifically Euclidean space, generalizing the rules of Euclidean space. Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular, a vector with four components can be used to represent a position in four-dimensional space. Spacetime is not a Euclidean space. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, one of time. The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. In 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams. But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, so developed along quite different lines. Minkowski's geometry of space-time is not Euclidean, consequently has no connection with the present investigation.
Four-dimensional space
–
5-cell
Four-dimensional space
–
3D projection of a tesseract undergoing a simple rotation in four dimensional space.
61.
Tesseract
–
In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes. The tesseract tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of "measure polytopes". In this publication, well as some of Hinton's later work, the word was occasionally spelled "tessaract". The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named with symmetry order 96. As a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64. As an orthotope it can be represented with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the convex hull of the points. That is, it consists of the points: A tesseract is bounded by eight hyperplanes.
Tesseract
–
Schlegel diagram
62.
List of geometers
List of geometers
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
List of geometers
–
Pythagoras
List of geometers
–
Euclid
List of geometers
–
Archimedes
63.
Yasuaki Aida
–
Aida Yasuaki also known as Aida Ammei, was a Japanese mathematician in the Edo period. He furthered methods for simplifying continued fractions. Aida created an original symbol for "equal". This was the first appearance in East Asia. History of mathematics in Japan. Tōkyō: _____. OCLC 122770600 Restivo, Sal P.. Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 978-0-7923-1765-4; OCLC 25709270 Selin, Helaine. . Encyclopaedia of the History of Science, Technology, Medicine in Non-Western Cultures. Dordrecht: Kluwer/Springer. ISBN 978-0-7923-4066-9; OCLC 186451909 Shimodaira, Kazuo. .
Yasuaki Aida
–
Aida Yasuaki
64.
Aryabhata
–
Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Arya-siddhanta. Furthermore, in most instances "Aryabhatta" would not fit the metre either. It mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This implies that he was born in 476. It called a native of Kusumapura or Pataliputra. Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in central India. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr; however, old records show that the city was actually Koṭum-kol-ūr. K. Chandra Hari has argued on the basis of astronomical evidence. Aryabhata is fairly certain that, at some point, he lived there for some time. Both Hindu and Buddhist tradition, well as Bhāskara I, identify Kusumapura as Pāṭaliputra, modern Patna. It is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. It is the author of several treatises on astronomy, some of which are lost. His major work, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times.
Aryabhata
–
Statue of Aryabhata on the grounds of IUCAA, Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.
Aryabhata
–
India's first satellite named after Aryabhata
65.
Ahmes
Ahmes
–
A portion of the Rhind Mathematical Papyrus
66.
Alhazen
–
Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, visual perception. Alhazen was the first to explain that vision occurs when light bounces on then is directed to one's eyes. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called "The Physicist". Alhazen is also sometimes called al-Baṣrī after his Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born 965 in Basra, then part of the Buyid emirate, to an Arab family. He arrived under the reign of Fatimid Caliph al-Hakim, a patron of the sciences, particularly interested in astronomy. He continued to live in Cairo, until his death in 1040. During this time, Alhazen continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. He made significant contributions to optics, number theory, geometry, natural philosophy. Alhazen's work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the eleventh-century prince of author of an important mathematical text, al-Mu ` taman ibn Hūd. A Latin translation of the Kitab al-Manazir was made probably in early thirteenth century. His research in catoptrics centred on mirrors and spherical aberration. Alhazen made the observation that the ratio between the angle of refraction does not remain constant, investigated the magnifying power of a lens.
Alhazen
–
Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Alhazen
–
Alhazen (Ibn al-Haytham)
Alhazen
–
The theorem of Ibn Haytham
Alhazen
–
Alhazen on Iraqi 10 dinars
67.
Apollonius of Perga
–
Apollonius of Perga was a Greek geometer and astronomer noted for his writings on conic sections. His innovative terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Johannes Kepler, Isaac Newton, René Descartes. Apollonius gave the ellipse, the hyperbola their modern names. Ptolemy describes Apollonius' theorem in the Almagest XII.1. Apollonius also researched the history, for which he is said to have been called Epsilon. The crater Apollonius on the Moon is named in his honor. He is one of the ancient geometers. The degree of originality of the Conics can best be judged from Apollonius's own prefaces. Books i–iv he describes as an "elementary introduction" containing essential principles, while the other books are specialized investigations in particular directions. Allusions such as Euclid's four Books on Conics, show a debt not only to Euclid but also to Conon and Nicoteles. The way the cone is cut does not matter. It is the form of the fundamental property that leads him to give their names: parabola, ellipse, hyperbola. Thus Books v–vii are clearly original. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations.
Apollonius of Perga
–
Pages from the 9th century Arabic translation of the Conics
Apollonius of Perga
–
Parabola connection with areas of a square and a rectangle, that inspired Apollonius of Perga to give the parabola its current name.
68.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed despite orders that he should not be harmed. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. The date of birth is based by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to the ruler of Syracuse. This work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever had children. During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred as his friend while two of his works have introductions addressed to Eratosthenes. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. He declined, saying that he had to finish working on the problem.
Archimedes
–
Archimedes Thoughtful by Fetti (1620)
Archimedes
–
Cicero Discovering the Tomb of Archimedes by Benjamin West (1805)
Archimedes
–
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi.
Archimedes
–
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A sphere and cylinder were placed on the tomb of Archimedes at his request. (see also: Equiareal map)
69.
Michael Atiyah
–
Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng is an English mathematician specialising in geometry. Since 1997, Atiyah has been an honorary professor at the University of Edinburgh. His students include Graeme Segal, Nigel Hitchin and Simon Donaldson. Atiyah was awarded the Fields Medal in 1966, the Abel Prize in 2004. He was born to a Lebanese father, the academic, Eastern Orthodox, Edward Atiyah and Scot Jean Atiyah. Patrick Atiyah is his brother; he has one other brother, a sister, Selma. Atiyah did his national service with the Royal Electrical and Mechanical Engineers. His postgraduate studies took place at Trinity College, Cambridge. He married July 1955 with whom he has three sons. In 1961, Atiyah moved to the University of Oxford, where he was a reader and fellow at St Catherine's College. Atiyah became a professorial fellow of New College, Oxford, from 1963 to 1969. Atiyah was president of the London Mathematical Society from 1974 to 1976. He has been active on the international scene, from 1997 to 2002. Atiyah also contributed on International Issues, the Association of European Academies, the European Mathematical Society. Within the United Kingdom, Atiyah was its first director.
Michael Atiyah
–
Michael Atiyah in 2007.
Michael Atiyah
–
Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master
Michael Atiyah
–
The Institute for Advanced Study in Princeton, where Atiyah was professor from 1969 to 1972
Michael Atiyah
–
The Mathematical Institute in Oxford, where Atiyah supervised many of his students
70.
Brahmagupta
–
Brahmagupta was an Indian mathematician and astronomer. Brahmagupta is the author of two early works on mathematics and astronomy: the Khaṇḍakhādyaka, a more practical text. According to his commentators, he was a native of Bhinmal. He was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. He was born according to his own statement. Brahmagupta lived during the reign of the Chapa dynasty ruler Vyagrahamukha. Brahmagupta was the son of Jishnugupta. Brahmagupta was a Shaivite by religion. Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, Brahmagupta worked there for a good part of his life. A later commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Abu region. It was also a center of learning for astronomy.
Brahmagupta
71.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was spent most of his adult life in Canada. In his youth, Coxeter was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in the Canadian Music Journal. He published twelve books. He was most noted for his work on higher-dimensional geometries. He was a champion of the classical approach in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Cambridge in 1926 to read mathematics. There he earned his doctorate in 1931. In 1932 he went as a Rockefeller Fellow where he worked with Hermann Weyl, Oswald Veblen, Solomon Lefschetz. Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P.
Harold Scott MacDonald Coxeter
–
Harold Scott MacDonald Coxeter
72.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, rigor. Euclid is the anglicized version of the Greek Εὐκλείδης, which means "renowned, glorious". Very original references to Euclid survive, so little is known about his life. The place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is usually referred to as" ὁ στοιχειώτης". The historical references to Euclid were written centuries after he lived by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements. This anecdote is questionable since it is similar to a story told about Alexander the Great. 247–222 BC. A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a town of Tyre. This biography is generally believed to be completely fictitious. However, there is little evidence in its favor.
Euclid
–
Euclid by Justus van Gent, 15th century
Euclid
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
–
Statue in honor of Euclid in the Oxford University Museum of Natural History
73.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Euler is also known for his work in mechanics, music theory. Euler was one of the most eminent mathematicians of the 18th century, is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field. He spent most of his adult life in St. Petersburg, Russia, in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times.
Leonhard Euler
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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
–
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
74.
Carl Friedrich Gauss
–
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, as the son of poor working-class parents. He was confirmed in a church near the school he attended as a child. Gauss was a prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. He made his first ground-breaking mathematical discoveries while still a teenager. He completed his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work has shaped the field to the present day. While at university, Gauss independently rediscovered important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in theory. On April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.
Carl Friedrich Gauss
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Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Carl Friedrich Gauss
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Statue of Gauss at his birthplace, Brunswick
Carl Friedrich Gauss
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Title page of Gauss's Disquisitiones Arithmeticae
Carl Friedrich Gauss
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Gauss's portrait published in Astronomische Nachrichten 1828
75.
Mikhail Leonidovich Gromov
–
Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of a Professor of Mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Jewish mother Lea Rabinovitz were pathologists. Gromov studied mathematics at Leningrad State University where he defended his Postdoctoral Thesis in 1973. His advisor was Vladimir Rokhlin. Gromov married in 1967. In 1970, invited to give a presentation in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings. Disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel. He changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged at Stony Brook. He adopted French citizenship in 1992.
Mikhail Leonidovich Gromov
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Mikhail Gromov
76.
David Hilbert
–
David Hilbert was a German mathematician. He is recognized as one of universal mathematicians of the 19th and early 20th centuries. Hilbert developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of the foundations of functional analysis. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. His students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium; but, after an unhappy period, he graduated from the more science-oriented Wilhelm Gymnasium. In autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In Hermann Minkowski, returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the gifted Minkowski". In 1884, Adolf Hurwitz arrived as an Extraordinarius. Hilbert obtained his doctorate with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg from 1886 to 1895. As a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen.
David Hilbert
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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
77.
Felix Klein
–
His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Klein's mother was Sophie Elise Klein. He attended the Gymnasium in Düsseldorf, intending to become a physicist. Klein received his doctorate, supervised by Plücker, in 1868. Plücker died in 1868, leaving his book on the foundations of geometry incomplete. Klein visited the following year along with visits to Berlin and Paris. At the outbreak of the Franco-Prussian War, he was in Paris and had to leave the country. For a short time, he served before being appointed lecturer at Göttingen in early 1871. Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule, Klein was appointed at Leipzig. There his colleagues included Walther von Dyck, Rohn, Friedrich Engel. 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883–1884, he was plagued by depression.
Felix Klein
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Felix Klein
78.
Nikolai Lobachevsky
–
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. He was one of three children. His father, a clerk in a land surveying office, died when he was seven, his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, founded just three years earlier in 1804. At Kazan University, he was influenced by friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a master's degree in physics and mathematics in 1811. He served in many administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children. He was dismissed due to his deteriorating health: by the early 1850s, he was nearly unable to walk. He died in poverty in 1856. He was an atheist. Lobachevsky's main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid's fifth postulate from other axioms.
Nikolai Lobachevsky
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Portrait by Lev Kryukov (c. 1843)
Nikolai Lobachevsky
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Annual celebration of Lobachevsky's birthday by participants of Volga 's student Mathematical Olympiad
79.
Hermann Minkowski
–
Hermann Minkowski was a Jewish German mathematician, professor at Königsberg, Zürich and Göttingen. He used geometrical methods to solve problems in number theory, mathematical physics, the theory of relativity. Hermann was a younger brother of Oskar. In different sources Minkowski's nationality is variously given as German, Polish, Lithuanian or Lithuanian-German, or Russian. Minkowski taught in Bonn, Königsberg and Zürich, finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; inventor Reinhold Rudenberg was his son-in-law. Minkowski died suddenly on 12 January 1909. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death can not take away is his noble image in the knowledge that his spirit continues to be active in us. The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor. Minkowski was educated at the Albertina University of Königsberg where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. He also became a friend of David Hilbert. Oskar Minkowski, was a well-known physician and researcher.
Hermann Minkowski
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Hermann Minkowski
80.
Minggatu
–
Minggatu, full name Sharabiin Myangat was a Mongolian astronomer, mathematician, topographic scientist at the Qing court. His courtesy name was Jing An. Minggatu was born in Plain White Banner of the Qing Empire. He was of the Sharaid clan. His name first appeared as a shengyuan of the Imperial Astronomical Bureau. He worked there at a time when Jesuit missionaries were in charge of calendar reforms. He also joined the team of China's measurement. From 1724 up to 1759, he worked at the Imperial Observatory. He participated in editing the study of the armillary sphere. He was the first person in China who calculated infinite series and obtained more than 10 formulae. In the 1730s, he first established and used what was later to be known as Catalan numbers. Minggatu's work is remarkable in that expansions in logarithmic were apprehended algebraically and inductively without the aid of differential and integral calculus. In 1742 he participated in the revision of the Compendium of Observational and Computational Astronomy. In 1756, he participated in the surveying of the Dzungar Khanate, incorporated into the Qing Empire by the Qianlong Emperor. It was due to his geographical surveys in Xinjiang that the Complete Atlas of the Empire was finished.
Minggatu
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Minggatu
Minggatu
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A page from Ming Antu's Geyuan Milv Jifa
Minggatu
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Ming Antu's geometrical model for trigonometric infinite series
Minggatu
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Ming Antu discovered Catalan numbers
81.
Blaise Pascal
–
Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy, educated by his father, a tax collector in Rouen. Pascal also wrote in defence of the scientific method. In 1642, while still a teenager, he started some pioneering work on calculating machines. Following Galileo Galilei and Torricelli, in 1646, he rebutted Aristotle's followers who insisted that nature abhors a vacuum. Pascal's results caused many disputes before being accepted. In 1646, his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. His father died in 1651. Following a religious experience in late 1654, he began writing influential works on theology. His two most famous works date from this period: the Lettres provinciales and the Pensées, the former set in the conflict between Jansenists and Jesuits. In that year, he also wrote an important treatise on the arithmetical triangle. Between 1659 he wrote on the cycloid and its use in calculating the volume of solids. He died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, in France's Auvergne region. He lost Antoinette Begon, at the age of three.
Blaise Pascal
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Painting of Blaise Pascal made by François II Quesnel for Gérard Edelinck in 1691.
Blaise Pascal
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An early Pascaline on display at the Musée des Arts et Métiers, Paris
Blaise Pascal
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Portrait of Pascal
Blaise Pascal
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Pascal studying the cycloid, by Augustin Pajou, 1785, Louvre
82.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so little reliable information is known about him. He travelled, visiting Egypt and Greece, maybe India. Around 530 BC, there established some kind of school or guild. In 520 BC, he returned to Samos. Pythagoras made influential contributions in the late 6th century BC. He is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his successors. Some accounts mention that numbers were important. Burkert states that Aristoxenus and Dicaearchus are the most important accounts. Aristotle had written a separate work On the Pythagoreans, no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans. Dicaearchus, Aristoxenus, Heraclides Ponticus had written on the same subject. According to Clement of Alexandria, Pythagoras was a disciple of Soches, Plato of Sechnuphis of Heliopolis. Herodotus, other early writers agree that Pythagoras was the son of Mnesarchus, born on a Greek island in the eastern Aegean called Samos.
Pythagoras
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Bust of Pythagoras of Samos in the Capitoline Museums, Rome.
Pythagoras
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Bust of Pythagoras, Vatican
Pythagoras
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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
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Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
83.
Bernhard Riemann
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Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, differential geometry. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a geometric treatment of complex analysis. Through his pioneering contributions to geometry, he laid the foundations of the mathematics of general relativity. Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. Charlotte Ebell, died before her children had reached adulthood. He was the second of six children, shy and suffering from nervous breakdowns. He suffered from timidity and a fear of speaking in public. During 1840, he went to Hanover to attend lyceum. After the death of his grandmother in 1842, Riemann attended high school at the Johanneum Lüneburg. He was often distracted by mathematics. His teachers were amazed by his adept ability to perform mathematical operations, in which he often outstripped his instructor's knowledge. At the age of 19, Riemann started studying philology and Christian theology in order to become a pastor and help with his family's finances. Once there, Riemann began studying mathematics under Carl Friedrich Gauss. During his time of study, Jacobi, Lejeune Dirichlet, Eisenstein were teaching. Riemann returned to Göttingen in 1849.
Bernhard Riemann
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Bernhard Riemann in 1863.
Bernhard Riemann
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Riemann's tombstone in Biganzolo
84.
Sijzi
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Abu Sa'id Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi was an Iranian Muslim astronomer, mathematician, astrologer. He is notable for proposing that the Earth rotates around its axis in the 10th century. He dedicated work to'Adud al-Daula, probably his patron, to the prince of Balkh. He also worked in Shiraz making astronomical observations from 969 to 970. Al-Sijzi studied intersections of conic circles. By my life, it is a problem difficult of refutation. For it is the same whether you take it that the Earth is in the sky. For, in both cases, it does not affect the Astronomical Science. It is just for the physicist to see if it is possible to refute it. Al-Biruni also referred as a prominent astronomer who defended the theory that the earth rotates in al-Qānūn al-Masʿūdī. O'Connor, John J.; Robertson, Edmund F. "Abu Said Ahmad ibn Muhammad Al-Sijzi", MacTutor History of Mathematics archive, University of St Andrews. Hogendijk, Jan P.. Al-Sijzi's Treatise on Geometrical Problem Solving. Tehran: Fatemi Publishing Co. ISBN 964-318-114-6.
Sijzi
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A page from Al Sijzi's geometrical treatise.
85.
Nasir al-Din al-Tusi
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Khawaja Muhammad ibn Muhammad ibn Hasan Tūsī, better known as Nasīr al-Dīn Tūsī, was a Persian polymath, architect, philosopher, physician, scientist, theologian and Marja Taqleed. He was of the Twelver Shī‘ah Islamic belief. The Muslim scholar Ibn Khaldun considered Tusi to be the greatest of the later Persian scholars. Nasir al-Din Tusi was born in the city of Tus in medieval Khorasan in the year 1201 and began his studies at an early age. In Hamadan and Tus he studied the Qur ` an, Hadith, Shi'a jurisprudence, logic, philosophy, astronomy. He was apparently born into a Shī‘ah family and lost his father at a young age. At a young age he moved to Nishapur to study philosophy under Farid al-Din Damad and mathematics under Muhammad Hasib. He met also Farid al-Din'Attar, the legendary Sufi master, later killed by Mongol invaders, he attended the lectures of Qutb al-Din al-Misri. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus. He was captured after the invasion of the Alamut castle by the Mongol forces. Here are some of his major works: Kitāb al-Shakl al-qattāʴ Book on the complete quadrilateral. A five volume summary of trigonometry. Al-Tadhkirah fi'ilm al-hay'ah – A memoir on the science of astronomy. Many commentaries were written about this work called Sharh al-Tadhkirah - Commentaries were written by Abd al-Ali ibn Muhammad ibn al-Husayn al-Birjandi and by Nazzam Nishapuri. Akhlaq-i Nasiri – A work on ethics.
Nasir al-Din al-Tusi
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Persian Muslim scholar Nasīr al-Dīn Tūsī
Nasir al-Din al-Tusi
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A Treatise on Astrolabe by Tusi, Isfahan 1505
Nasir al-Din al-Tusi
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Tusi couple from Vat. Arabic ms 319
Nasir al-Din al-Tusi
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The Astronomical Observatory of Nasir al- Dīn Tusi.
86.
Oswald Veblen
–
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan theorem in 1905; while this was long considered the first rigorous proof, many now also consider Jordan's original proof rigorous. Veblen was born in Decorah, Iowa. His parents were Kirsti Veblen. Veblen's uncle was Thorstein Veblen, sociologist. He went in Iowa City. For his graduate studies, he went to study mathematics at the University of Chicago, where he obtained a Ph.D. in 1903. A System of Axioms for Geometry was written under the supervision of E. H. Moore. During World War I, Veblen served first as a captain, later as a major in the army. Veblen taught mathematics to 1932. In 1926, he was named Henry B. Fine Professor of Mathematics. In 1932, he helped resigning his professorship to become the first professor at the Institute that same year. He kept his professorship at the Institute until he was made emeritus in 1950. In Princeton Veblen and his wife Elizabeth accumulated land along the Princeton Ridge.
Oswald Veblen
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Oswald Veblen (photo ca. 1915)
87.
Yang Hui
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Yang Hui, courtesy name Qianguang, was a late-Song dynasty Chinese mathematician from Qiantang. Yang is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the famous mathematician Qin Jiushao. In his book known as Piling-up Powers and Unlocking Coefficients, known through his contemporary mathematician Liu Ruxie. Jia described the method used as ` li cheng suo'. It appeared again in a publication of Zhu Shijie's book Jade Mirror of the Four Unknowns of 1303 AD. Around 1275 AD, Yang finally had two mathematical books, which were known as the Xugu Zhaiqi Suanfa and the Suanfa Tongbian Benmo. There were also theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system. History of mathematics List of mathematicians Chinese mathematics Needham, Joseph. Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Li, Jimin, "Yang Hui". Encyclopedia of China, 1st ed. Yang Hui at MacTutor
Yang Hui
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1433 Korean edition of Yang Hui suan fa
Yang Hui
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Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD.
88.
Zhang Heng
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Zhang Heng, formerly romanized as Chang Heng, was a Han Chinese polymath from Nanyang who lived during the Han dynasty. Zhang Heng began his career in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors at the imperial court. His uncompromising stance on calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. Zhang returned home to Nanyang before being recalled to serve in the capital once more in 138. He died a year later in 139. Zhang applied his extensive knowledge of gears in several of his inventions. He improved Chinese calculations for pi. His fu and poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many posthumous honors for his ingenuity; some modern scholars have compared his work in astronomy to that of the Greco-Roman Ptolemy. Born in Nanyang Commandery, Zhang Heng came from a distinguished but not very affluent family. At age ten, Zhang's father died, leaving him in the care of his grandmother. An accomplished writer in Zhang left home in the year 95 to pursue his studies in the capitals of Chang ` an and Luoyang. While traveling to Luoyang, Zhang dedicated one of his earliest fu poems to it. He acted modestly and declined.
Zhang Heng
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A stamp of Zhang Heng issued by China Post in 1955
Zhang Heng
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A 2nd-century lacquer-painted scene on a basket box showing famous figures from Chinese history who were paragons of filial piety: Zhang Heng became well-versed at an early age in the Chinese classics and the philosophy of China's earlier sages.
Zhang Heng
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A Western Han terracotta cavalier figurine wearing robes and a hat. As Chief Astronomer, Zhang Heng earned a fixed salary and rank of 600 bushels of grain (which was mostly commuted to payments in coinage currency or bolts of silk), and so he would have worn a specified type of robe, ridden in a specified type of carriage, and held a unique emblem that marked his status in the official hierarchy.
Zhang Heng
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A pottery miniature of a palace made during the Han Dynasty; as a palace attendant, Zhang Heng had personal access to Emperor Shun and the right to escort him
89.
Before Common Era
–
Common Era or Current Era, abbreviated CE, is a calendar era, often used as an alternative naming of the Anno Domini era, abbreviated AD. The system uses BCE as an abbreviation for "before the Common Era" and CE as an abbreviation for "Common Era". The year-numbering system associated with it is the calendar system with most widespread use in the world today. For decades, it has been the global standard, recognized by international institutions such as the Universal Postal Union. At those times, the expressions were all used interchangeably with "Christian Era". Use of the CE abbreviation was introduced by Jewish academics in the mid-19th century. He attempted from an initial epoch, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the Easter table in which he introduced the new era "Anni Domini Nostri Jesu Christi". Numbering years in this manner became more widespread with its usage in England in 731. In 1422, Portugal became the last European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered far was in a 1615 book by Johannes Kepler. Kepler uses it again in a 1616 table of ephemerides, again in 1617. A 1635 English edition of that book has the page in English -- so far, the earliest-found usage of Vulgar Era in English. A 1701 book edited by John LeClerc includes "Before Christ according to 6". A 1796 book uses the term "era of the nativity".
Before Common Era
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Key concepts
90.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, change. There is a range of views among philosophers as to the exact scope and definition of mathematics. Mathematicians use them to formulate new conjectures. Mathematicians resolve the falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of logic, mathematics developed from counting, calculation, measurement, the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Galileo Galilei said, "The universe can not become familiar with the characters in which it is written. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss referred as "the Queen of the Sciences". Benjamin Peirce called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians
91.
Right triangle
–
A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the angles of a right triangle is the basis for trigonometry. The right angle is called the hypotenuse. The sides adjacent to the right angle are called legs. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. As a formula the T is T = 1 2 a b where a and b are the legs of the triangle. This formula only applies to right triangles. From this: The altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse. Each leg of the triangle is the mean proportional of the segment of the hypotenuse, adjacent to the leg. Thus f = a b c. For solutions of this equation in integer values of a, b, f, c, see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the triangle's orthocenter -- the intersection of its three altitudes -- coincides with the right-angled vertex. Pythagorean triples are integer values of b, c satisfying this equation. The radius of the circumcircle is half the length of the hypotenuse, R = 2.
Right triangle
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Right triangle
92.
Right angle
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In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two adjacent parts of a straight line. More precisely, if a ray is placed so that the adjacent angles are equal, then they are right angles. As a rotation, a right angle corresponds to a turn. The presence of a right angle in a triangle is the defining factor for right triangles, making the right basic to trigonometry. The term is a calque of Latin rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. In Unicode, the symbol for a right angle is U+221F ∟ RIGHT ANGLE. It should not be confused with the similarly shaped symbol U+231E ⌞ BOTTOM LEFT CORNER. Related symbols are U +22 BE ⊾ RIGHT ANGLE WITH ARC, U +299 D ⦝ MEASURED RIGHT ANGLE WITH DOT. Right angles are fundamental in Euclid's Elements. They are defined in definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 12 to define acute angles and obtuse angles. Two angles are called complementary if their sum is a right angle. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. A right angle may be expressed in different units: 1/4 turn.
Right angle
–
A right angle is equal to 90 degrees.
93.
Cathetus
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In a right triangle, a cathetus, commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a "side about the right angle". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as "the other two sides". If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor and major cathetus. By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse. Bernhardsen, T. Geographic Information Systems: An Introduction, 3rd ed. New York: Wiley, p. 271, 2002. Cathetus at Encyclopaedia of Mathematics Weisstein, Eric W. "Cathetus". MathWorld.
Cathetus
–
A right-angled triangle where c 1 and c 2 are the catheti and h is the hypotenuse
94.
Theorem
–
A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the statement. Mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. However, the conditional could be interpreted differently depending on the meanings assigned to the derivation rules and the conditional symbol. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". Its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically, many theorems are of the form of an conditional: if A, then B. Such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion. To be proved, a theorem must be expressible as a formal statement.
Theorem
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A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
95.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: conditional equations. An equation is true for all values of the variable. A conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation. Each member will contain one or more terms. A x 2 + B x + C = y has two members: A x 2 + B x + C and y. The left member has the right member one term. The parameters are A, B, C. An equation is analogous to a scale into which weights are placed. In geometry, equations are used to describe geometric figures. This is the starting idea of an important area of mathematics. Algebra studies two main families of equations: polynomial equations and, among them the special case of linear equations.
Equation
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A strange attractor which arises when solving a certain differential equation.
96.
Ancient Greece
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Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages to c. 5th century BC to the end of antiquity. Immediately following this period was the beginning of the Byzantine era. Included in ancient Greece is the period of Classical Greece, which flourished during the 5th to 4th centuries BC. Classical Greece began with the era of the Persian Wars. Because of conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished to the western end of the Mediterranean Sea. Classical Antiquity in the Mediterranean region is commonly considered to have ended in the 6th century AD. Classical Antiquity in Greece is preceded by the Dark Ages, archaeologically characterised by the protogeometric and geometric styles of designs on pottery. The end of the Dark Ages is also frequently dated to the year of the first Olympic Games. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures with the dreamlike "archaic smile". The Archaic period is often taken to end in 508 BC. This period saw Greco-Persian Wars and the Rise of Macedon. Following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period ends with the Roman conquest. Herodotus is widely known as the "father of history": his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato and Aristotle.
Ancient Greece
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The Parthenon, a temple dedicated to Athena, located on the Acropolis in Athens, is one of the most representative symbols of the culture and sophistication of the ancient Greeks.
Ancient Greece
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Dipylon Vase of the late Geometric period, or the beginning of the Archaic period, c. 750 BC.
Ancient Greece
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Political geography of ancient Greece in the Archaic and Classical periods
97.
Mathematical proof
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In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. An unproved proposition, believed to be true is known as a conjecture. Proofs usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics. The philosophy of mathematics is concerned with mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren. The early use of "probity" was in the presentation of legal evidence. A person such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as analogies preceded strict mathematical proof.
Mathematical proof
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One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Mathematical proof
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Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
98.
Babylonian mathematics
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Babylonian mathematical texts are plentiful and well edited. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, baked hard in an oven or by the heat of the sun. The Babylonian tablet YBC 7289 gives an approximation to 2 accurate to three significant sexagesimal digits. Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. The Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values. The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC.
Babylonian mathematics
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Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
99.
Mesopotamia
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In the Iron Age, it was controlled by the Neo-Assyrian and Neo-Babylonian Empires. After his death, it became part of the Greek Seleucid Empire. Around 150 BC, Mesopotamia was under the control of the Parthian Empire. Mesopotamia became a battleground between the Romans and Parthians, with parts of Mesopotamia coming under Roman control. In AD 226, it remained under Persian rule until the 7th century Muslim conquest of Persia of the Sasanian Empire. A number of Christian native Mesopotamian states existed between the 1st century BC and 3rd century AD, including Adiabene, Osroene, Hatra. Mesopotamia is the site of the earliest developments of the Neolithic Revolution from around 10,000 BC. The regional toponym Mesopotamia comes from the Greek root words μέσος "middle" and ποταμός "river" and literally means" between two/the rivers". It is used throughout the Greek Septuagint to translate the Hebrew Naharaim. In the Anabasis, Mesopotamia was used to designate the land east of the Euphrates in north Syria. The Aramaic term narim corresponded to a similar geographical concept. The neighbouring steppes to the western part of the Zagros Mountains are also often included under the wider term Mesopotamia. A further distinction is usually made between Southern or Lower Mesopotamia. Upper Mesopotamia, also known as the Jazira, is the area from their sources down to Baghdad. Lower Mesopotamia consists of southern Iraq, parts of western Iran.
Mesopotamia
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Known world of the Mesopotamian, Babylonian, and Assyrian cultures from documentary sources
Mesopotamia
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Map showing the extent of Mesopotamia
Mesopotamia
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One of 18 Statues of Gudea, a ruler around 2090 BC
Mesopotamia
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One of the Nimrud ivories shows a lion eating a man. Neo-Assyrian period, 9th to 7th centuries BC.
100.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions as a number, algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. This was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They mass-produced weights in geometrical shapes, which included hexahedra, cylinders, thereby demonstrating knowledge of basic geometry. The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts.
Indian mathematics
Indian mathematics
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The design of the domestic fire altar in the Śulba Sūtra
101.
Chinese mathematics
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Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, trigonometry. Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, even after this date the manuscript traditions are obscure. Chinese mathematicians made advances in algorithm algebra. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as by Shen Kuo. Simple mathematics on Oracle bone script date back to the Shang Dynasty. One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty. For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching contained elements of binary numbers. Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood negative numbers with counting rods. Math was one of the Liù Yì or Six Arts, students were required to master during the Zhou Dynasty.
Chinese mathematics
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Visual proof for the (3, 4, 5) triangle as in the Zhou Bi Suan Jing 500–200 BC.
Chinese mathematics
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counting rod place value decimal
Chinese mathematics
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Lui Hui's Survey of sea island
Chinese mathematics
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Li Ye's inscribed circle in triangle: Diagram of a round town
102.
Q.E.D.
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Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, meaning ", what had to be shown" or "thus it has been demonstrated". The abbreviation thus signals the completion of the proof. The phrase quod demonstrandum is a translation from the Greek ὅπερ ἔδει δεῖξαι. Translating from the Latin into English yields, "what was to be demonstrated"; however, translating the Greek phrase ὅπερ ἔδει δεῖξαι produces a slightly different meaning. The phrase was used by Greek mathematicians, including Euclid and Archimedes. In the European Renaissance, scholars often wrote in Latin, phrases such as Q.E.D. were often used to conclude proofs. Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's magnum opus. The system of the book is, as Spinoza says, "demonstrated in geometrical order", with definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary. There is another Latin phrase with a slightly different meaning, less common in usage. Quod faciendum, originating from the Greek geometers' closing ἔδει ποιῆσαι, meaning "which had to be done". Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid's first proposition showing how to construct an equilateral triangle given one side is concluded this way. The phrase is usually shortened to QEF.
Q.E.D.
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Spinoza 's original text of Ethics, Part 1. Q.E.D. is used at the end of DEMONSTRATIO of PROPOSITIO III. in the right page.
103.
Law of cosines
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In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. This formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an entirely analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi. The theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form. It is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a the quadratic equation 2ab γ + b2 − c2 = 0. This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data. These different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c, where θ is the measurement of the angle opposite the side of length c. By the distance formula, we have c = 2 + 2. An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. right vs. obtuse.
Law of cosines
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Figure 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.
104.
Quadratic reciprocity
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In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem. This law, combined with the properties of the Legendre symbol, means that any Legendre symbol can be calculated. This makes it possible to determine, for x 2 ≡ a, where p is an odd prime, whether it has a solution. However, it does not provide any help at all for actually finding the solution. The solution can be found using quadratic residues. The theorem was first proved by Gauss. Privately he referred as the "golden theorem." He two more were found in his posthumous papers. There are now over 200 published proofs. The first section of this article gives a special case of quadratic reciprocity, representative of the general case. The second section gives the formulations of quadratic reciprocity found by Legendre and Gauss. Consider the polynomial f = n 2 − 5 and its values for n ∈ N. No primes ending in 3 or 7 ever appear. In other words, 5 is a quadratic residue p iff p is a quadratic residue modulo 5.
Quadratic reciprocity
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Part of Article 131 in the first edition (1801) of the Disquisitiones, listing the 8 cases of quadratic reciprocity
105.
Proportionality (mathematics)
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The constant is called the coefficient of proportionality or proportionality constant. If one variable is always the product of a constant, the two are said to be directly proportional. X and y are directly proportional if the y/x is constant. If the product of the two variables is always a constant, the two are said to be inversely proportional. X and y are inversely proportional if the xy is constant. To express the statement "y is directly proportional to x" where c is the proportionality constant. Symbolically, this is written as y ∝ x. To express the statement "y is inversely proportional to x" mathematically, we write an equation y = c/x. We can equivalently write "y is directly proportional to 1/x". An equality of two ratios is called a proportion. For example, a/c = b/d, where no term is zero. Given two variables x and y, y is directly proportional to x if there is a constant k such that y = k x. The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π. The concept of inverse proportionality can be contrasted against direct proportionality. Consider two variables said to be "inversely proportional" to each other.
Proportionality (mathematics)
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Variable y is directly proportional to the variable x.
106.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains six lemons, then the ratio of oranges to lemons is eight to six. Thus, a ratio can be a fraction as opposed to a whole number. Also, the ratio of oranges to the total amount of fruit is 8:14. The numbers compared in a ratio can be any quantities such as objects, persons, lengths, or spoonfuls. A ratio is written "a to b" or a:b, or sometimes expressed arithmetically as a quotient of the two. When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the ratio is often used instead for this more general notion as well. B being the consequent. The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. B and C are called the means. The equality of three or more proportions is called a continued proportion. Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four", ten inches long is 2:4:10.
Ratio
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The ratio of width to height of standard-definition television.
107.
Triangle postulate
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It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. It then must depend on the triangle. Its difference from ° is a case of angular defect and serves as an important distinction for geometric systems. In Euclidean geometry, the postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also. Equidistance postulate: Parallel lines are everywhere equidistant Triangle area property: The area of a triangle can be as large as we please. Three lie on a line or lie on a circle. Pythagoras' theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The sum of the angles of a hyperbolic triangle is less than 180°. The triangle's area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Proclus' axiom, the equidistance postulate, Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small.
Triangle postulate
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Equivalence of the parallel postulate and the "sum of the angles equals to 180°" statement
108.
Parallel postulate
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In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. Euclidean geometry is the study of geometry that satisfies all including the parallel postulate. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry, independent of Euclid's fifth postulate is known as absolute geometry. These equivalent statements include: There is at most one line that can be drawn parallel to another given one through an external point. The sum of the angles in every triangle is °. There exists a triangle whose angles add up to °. The sum of the angles is the same for every triangle. There exists a pair of not congruent, triangles. Every triangle can be circumscribed. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all angles are right angles, a rectangle. There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the same line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Parallel postulate
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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
109.
Cosine
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, tangent. More precise definitions are detailed below. Trigonometric functions have a wide range of uses including computing unknown angles in triangles. In this use, trigonometric functions are used, in navigation, engineering, physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six trigonometric functions, tabulated here with equations that relate them to one another. That is, for any similar triangle the ratio of another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express. To define the trigonometric functions for the A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows: The hypotenuse is the side in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
Cosine
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Trigonometric functions in the complex plane
Cosine
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Trigonometry
Cosine
Cosine
110.
Euclid's Elements
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Euclid's Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, mathematical proofs of the propositions. The books cover the ancient Greek version of elementary number theory. It is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. According to Proclus, the term "element" was used to describe a theorem that helps furnishing proofs of many other theorems. The element in the Greek language is the same as letter. This suggests that theorems in the Elements should be seen as standing as letters to language. Euclid's Elements has been referred to as the most influential textbook ever written. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, supplemented by some original work. The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated prior to Boethius in the fifth or sixth century. The Arabs received the Elements around 760; this version was translated into Arabic under Harun al Rashid circa 800. The Byzantine scholar Arethas commissioned the copying of the extant Greek manuscripts of Euclid in the late ninth century.
Euclid's Elements
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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Euclid's Elements
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A fragment of Euclid's "Elements" on part of the Oxyrhynchus papyri
Euclid's Elements
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An illumination from a manuscript based on Adelard of Bath 's translation of the Elements, c. 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclid's Elements
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Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
111.
Side angle side
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This means that either object can be reflected so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. In elementary geometry the congruent is often used as follows. The word equal is often used in place of congruent for these objects. Two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure. Two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size but not in shape. For two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have numerically identical sequences side-angle-side-angle... for n sides and n angles. Congruence of polygons can be established graphically as follows: First, label the corresponding vertices of the two figures. Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.
Side angle side
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An example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others. Note that congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distance and angles. The unchanged properties are called invariants.
112.
Albert Einstein
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Albert Einstein was a German-born theoretical physicist. Einstein developed the general theory of one of the two pillars of modern physics. Einstein's work is also known on the philosophy of science. Einstein is best known in popular culture for his mass -- energy equivalence E = mc2. This led him to develop his special theory of relativity. Einstein continued to deal with problems of statistical mechanics and theory, which led to his explanations of particle theory and the motion of molecules. Einstein also investigated the thermal properties of light which laid the foundation of the theory of light. In 1917, he applied the general theory of relativity to model the large-scale structure of the universe. Einstein settled in the U.S. becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project. He largely denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, he signed the Russell -- Einstein Manifesto, which highlighted the danger of nuclear weapons. He was affiliated with the Institute until his death in 1955. He published more than 300 scientific papers along over 150 non-scientific works. On 5 universities and archives announced the release of Einstein's papers, comprising more than 30,000 unique documents.
Albert Einstein
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Albert Einstein in 1921
Albert Einstein
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Einstein at the age of 3 in 1882
Albert Einstein
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Albert Einstein in 1893 (age 14)
Albert 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)
113.
James A. Garfield
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James Abram Garfield was the 20th President of the United States, serving from March 4, 1881, until his assassination later that year. He is the only sitting House member to be elected president. Garfield was raised by his widowed mother. He worked at various jobs, in his youth. Beginning at age 17, he attended several Ohio schools, then studied at Williams College in Williamstown, Massachusetts, from which he graduated in 1856. Garfield entered politics as a Republican. He served as a member of the Ohio State Senate. He was first elected in 1862 to represent Ohio's 19th District. Throughout Garfield's congressional service after the Civil War, he firmly supported the gold standard and gained a reputation as a skilled orator. Garfield initially later favored a moderate approach for civil rights enforcement for freedmen. In the 1880 presidential election, Garfield narrowly defeated Democrat Winfield Scott Hancock. Garfield made diplomatic judiciary appointments, including a U.S. Supreme Court justice. Garfield advocated civil rights for African Americans. James Garfield was born the youngest of five children on November 1831, in a log cabin in Orange Township, now Moreland Hills, Ohio. Like many who settled there, Garfield's ancestors were from New England.
James A. Garfield
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Brady - Handy photograph of Garfield, taken between 1870 and 1880
James A. Garfield
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Birthplace site of James Garfield
James A. Garfield
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Garfield at age 16
James A. Garfield
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Lucretia Garfield in the 1870s
114.
U.S. Representative
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The composition and powers of the House are established by Article One of the United States Constitution. Since its inception in 1789, all representatives are elected popularly. The total number of voting representatives is fixed by law at 435. The House is charged with the passage of federal legislation, known as bills, which, after concurrence by the Senate, are sent to the President for consideration. The presiding officer is the Speaker of the House, elected by the members thereof and is therefore traditionally the leader of the controlling party. Other floor leaders are chosen depending on whichever party has more voting members. The House meets in the south wing of the United States Capitol. All states except Rhode Island agreed to send delegates. The issue of how to structure Congress was one of the most divisive among the founders during the Convention. The House is referred to as the lower house, with the Senate being the upper house, although the United States Constitution does not use that terminology. Both houses' approval is necessary for the passage of legislation. The Virginia Plan drew the support of delegates from large states such as Virginia, Massachusetts, Pennsylvania, as it called for representation based on population. The smaller states, however, favored the New Jersey Plan, which called for a unicameral Congress with equal representation for the states. Its implementation was set for March 1789. The House began work on April 1, 1789, when it achieved a quorum for the first time.
U.S. Representative
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United States House of Representatives
U.S. Representative
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Seal of the House
U.S. Representative
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Republican Thomas Brackett Reed, occasionally ridiculed as "Czar Reed", was a U.S. Representative from Maine, and Speaker of the House from 1889 to 1891 and from 1895 to 1899.
U.S. Representative
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House Speaker Nancy Pelosi, Majority Leader Steny Hoyer, and Education and Labor Committee Chairman George Miller confer with President Barack Obama at the Oval Office in 2009.
115.
Calculus
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It has two major branches, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed by Isaac Newton and Gottfried Leibniz. Calculus has widespread uses in science, engineering and economics. Calculus is a part of modern education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". Calculus is also used for naming theories of computation, such as propositional calculus, calculus of variations, lambda calculus, process calculus. The method of exhaustion was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. The infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal term.
Calculus
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
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Maria Gaetana Agnesi
Calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
116.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, biology. In pure mathematics, differential equations mostly concerned with their solutions -- the set of functions that satisfy the equation. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence by Newton and Leibniz. Jacob Bernoulli proposed the Bernoulli equation in 1695. In 1746, within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. Lagrange sent the solution to Euler. Both further applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fourier's proposal of his equation for conductive diffusion of heat. This partial equation is now taught to every student of mathematical physics. In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one to express these variables dynamically as a equation for the unknown position of the body as a function of time.
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
117.
Triangle inequality
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The triangle inequality expresses a relationship between absolute values. The inequality can be viewed intuitively in either ℝ2 or ℝ3. The figure at the right shows three examples approaching equality. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. The inequality is a defining property of norms and measures of distance. Euclid proved the inequality for distances in plane geometry using the construction in the figure. It then is argued that angle β > α, so side AD > AC. But AD = AB + BD = AB + BC so the sum of sides AB + BC > AC. This proof appears in Book 1, Proposition 20. A more succinct form of this system can be shown to be | a − b | < c < a + b. A mathematically equivalent formulation is that c must be a real number. The second part of this theorem is already established for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right ADC. An isosceles triangle ABC is constructed with equal sides AB = AC.
Triangle inequality
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Three examples of the triangle inequality for triangles with sides of lengths x, y, z. The top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y.
118.
Edsger Dijkstra
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Edsger Wybe Dijkstra was a Dutch computer scientist. A theoretical physicist by training, Dijkstra worked from 1952 to 1962. Dijkstra was a professor of mathematics at a research fellow at the Burroughs Corporation. One of the most influential members of science's founding generation, he helped shape the new discipline from both an engineering and a theoretical perspective. Many of his papers are the source of new research areas. Several problems that are now standard in computer science were first identified by Dijkstra and/or bear names coined by him. During the 1970s this became the new programming orthodoxy. He was an theoretical pioneer in many research areas of computing science. A 1965 paper of his is credited with being the first paper in the field of concurrent programming. Dijkstra was also one of the early pioneers of the research on principles of distributed computing. Shortly before his death in 2002, Dijkstra received the ACM PODC Influential-Paper Award in distributed computing for his work on self-stabilization of computation. This annual award was renamed the following year in his honor. Edsger W. Dijkstra was born in Rotterdam. His father was a chemist, president of the Dutch Chemical Society; he was later its superintendent. His mother was a mathematician, but never had a formal job.
Edsger Dijkstra
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Edsger Wybe Dijkstra in 2002
Edsger Dijkstra
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The Eindhoven University of Technology, located in Eindhoven in the south of the Netherlands, where Dijkstra was a professor of mathematics from 1962 to 1984.
Edsger Dijkstra
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Vincent van Gogh and Edsger W. Dijkstra were among the most notable residents in Nuenen 's history.
Edsger Dijkstra
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The University of Texas at Austin, where Dijkstra held the Schlumberger Centennial Chair in Computer Sciences from 1984 until 1999.
119.
Sign function
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In mathematics, the sign function or signum function is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the function is often represented as sgn. Any real number can be expressed as its sign function: x = sgn ⋅ | x |. The numbers cancel and all we are left with is the sign of x. D | x | d x = sgn for x ≠ 0. The function is differentiable with derivative 0 everywhere except at 0. Using this identity, it is easy to derive the distributional derivative: d sgn d x = 2 d H d x = 2 δ. The signum can also be written using the Iverson notation: sgn = − +. For k ≫ 1, a smooth approximation of the function is sgn ≈ tanh. Another approximation is sgn ≈ x x 2 + ε 2. Which gets sharper as ε → 0; note that this is the derivative of x2 + ε2. See Heaviside step function – Analytic approximations. The function can be generalized to complex numbers as: sgn = z | z | for any complex number z except z = 0. The signum of a given complex number z is the point on the unit circle of the complex plane, nearest to z. Then, for z ≠ 0, sgn = e i arg z, where arg is the complex argument function.
Sign function
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Signum function y = sgn(x)
120.
Pythagorean triple
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A Pythagorean triple consists of three positive integers a, b, c, such that a2 + b2 = c2. Such a triple is commonly written, a well-known example is. If is a Pythagorean triple, then so is for any positive 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. Moreover, 2 do not have an common multiple because √ 2 is irrational. There are 16 primitive Pythagorean triples with c ≤ 100: Note, for example, not a primitive Pythagorean triple, as it is a multiple of. Each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The triple generated by Euclid's formula is only if n not both odd. Every primitive triple arises from a unique pair of coprime numbers m, n, one of, even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of a, b and c to m and n from Euclid's formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, cannot be generated using integer m and n. This can be remedied by inserting an additional parameter k to the formula.
Pythagorean triple
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The Pythagorean theorem: a 2 + b 2 = c 2
121.
Coprime
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That is, the only positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1. The denominator of a reduced fraction are coprime. In addition to gcd the notation a ⊥ b is sometimes used to indicate that a and b are relatively prime. 14 and 21 are not, because they are both divisible by 7. They are the only integers to be coprime with 0. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. Between 1 and n, is given by Euler's totient function φ. A set of integers can also be called coprime if its elements share no positive factor except 1. A set of integers is said to be coprime if a and b are coprime for every pair of different integers in it. A number of conditions are individually equivalent to b being coprime: No prime number divides both a and b. There exist integers x and y such that + by = 1. The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1. In other words, b is a unit in the Z/aZ of integers modulo a. The least common multiple of b is equal to their product ab, i.e. LCM = ab.
Coprime
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Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 x 9 lattice does not intersect any other lattice points
122.
Greatest common divisor
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For example, the GCD of 8 and 12 is 4. The greatest common divisor is also known as the greatest common factor, highest common divisor. This notion can be extended to polynomials and other commutative rings. In this article we will denote the greatest common divisor of two integers a and b as gcd. Some textbooks use. What is the greatest common divisor of 54 and 24? Thus the divisors of 54 are: 1, 2, 3, 6, 9, 54. Similarly, the divisors of 24 are: 1, 2, 3, 4, 6, 24. The numbers that these two lists share in common are the common divisors of 54 and 24: 1, 2, 3, 6. The greatest of these is 6. That is, the greatest common divisor of 54 and 24. One writes: gcd = 6. The greatest common divisor is useful for reducing fractions to be in lowest terms. For gcd = 14, therefore, 56 = 3 ⋅ 14 4 ⋅ 14 = 3 4. Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1.
Greatest common divisor
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Contents
123.
Spiral of Theodorus
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In geometry, the spiral of Theodorus is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene. The spiral is started with each leg having unit length. Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theaetetus split the rational numbers and irrational numbers into different categories. Each of the triangles' hypotenuses hi gives the square root of the natural number, with h1 = √ 2. Plato, tutored by Theodorus, questioned why Theodorus stopped at √17. The reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure. In 1958, Erich Teuffel proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of length are extended into a line, they will never pass through any of the other vertices of the total figure. Theodorus stopped his spiral with a hypotenuse of √ 17. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found. If φn is the angle of the nth triangle, then: tan = 1 n. Therefore, the growth of the angle φn of the next triangle n is: φ n = arctan . The sum of the angles of the first k triangles is called the total φ for the kth triangle.
Spiral of Theodorus
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The spiral of Theodorus up to the triangle with a hypotenuse of √17
124.
Rational number
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Since q may be equal to 1, every integer is a rational number. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true just for base 10, but also for any other integer base. A real number, not rational is called irrational. Irrational numbers include √ 2, π, φ. The decimal expansion of an irrational number continues without repeating. Since the set of real numbers is uncountable, almost all real numbers are irrational. In abstract algebra, the rational numbers together with certain operations of multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. The algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed by completion using Cauchy sequences, Dedekind cuts, or infinite decimals. Zero divided by any other integer equals zero; therefore, zero is a rational number. The term rational to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun "rational number".
Rational number
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A diagram showing a representation of the equivalent classes of pairs of integers
125.
Compass and straightedge constructions
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. The ancient Greek mathematicians first conceived compass-and-straightedge constructions, a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass and straightedge constructions
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A compass
Compass and straightedge constructions
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Creating a regular hexagon with a ruler and compass
126.
Square root
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For example, 4 and 4 are square roots of 16 because 42 = 2 = 16. For example, the square root of 9 is 3, denoted √ 9 = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the expression underneath the radical sign, in this example 9. A has two square roots: √ a, positive, − √ a, negative. Together, these two roots are denoted ± √a. For positive a, the principal square root can also be written in exponent notation, as a1/2. Square roots of negative numbers can be discussed within the framework of complex numbers. A method for finding very good approximations to the square roots of 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the square root of numbers having many digits. This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to 380 BC. The particular case √ 2 is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with length 1. A 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist. A symbol for square roots, written as an elaborate R, was invented by Regiomontanus.
Square root
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First leaf of the complex square root
Square root
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The mathematical expression 'The (principal) square root of x"
127.
Hippasus
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Hippasus of Metapontum, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, believed to have been discovered around the time that he lived. Little is known about the life of Hippasus. He may have lived in the late 5th century BC, about a century after the time of Pythagoras. Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans. Memory was the most valued faculty. According to one statement, Hippasus left no writings, according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute. Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is confused. Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, that the member who first divulged the secret perished by drowning. Iamblichus gives a series of inconsistent reports. Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour.
Hippasus
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Hippasus of Metapontum
128.
Complex number
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In this expression, b is the imaginary part of the complex number. A + bi can be identified with the point in the complex plane. As as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation = − 9 has no real solution, since the square of a real number can not be negative. Complex numbers provide a solution to this problem. According to the fundamental theorem of algebra, all polynomial equations with complex coefficients in a single variable have a solution in complex numbers. For example, 3.5 + 2i is a complex number. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part. For example, Re = − 3.5 Im = 2. Hence, in imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is sometimes known as the Cartesian form of z. A can be regarded as a complex number a + 0i whose imaginary part is 0.
Complex number
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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i 2 = −1.
129.
Absolute value
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In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, | x | = x | 0 | = 0. For example, the absolute value of − 3 is also 3. The absolute value of a number may be thought as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, vector spaces. The absolute value is closely related to the notions of magnitude, norm in various mathematical and physical contexts. The term absolute value has been used from at least 1806 in French and 1857 in English. The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical magnitude. The same notation is used when x is a set to denote cardinality; the meaning depends on context. As can be seen from the above definition, the absolute value of x is always never negative. Indeed, the notion of an abstract function in mathematics can be seen to be a generalisation of the absolute value of the difference. For example: Absolute value is used to define the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value can not be directly generalised for a complex number.
Absolute value
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The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.
130.
Complex plane
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In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. In particular, multiplication by a complex number of modulus 1 acts as a rotation. Geometric plots in the plane as Argand diagrams. These are named after Jean-Robert Argand, although they were first described by mathematician Caspar Wessel. Argand diagrams are frequently used to plot the positions of the zeroes of a function in the complex plane. In this customary notation the complex z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented as = =. Thus, if θ is one value of arg, the other values are given by arg = + 2nπ, where n is any integer ≠ 0. The theory of integration comprises a major part of complex analysis. By convention the positive direction is counterclockwise. It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a point in the plane, draw a straight line connecting it with the pole on the sphere. That line will intersect the surface of the sphere in exactly one other point.
Complex plane
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Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.
131.
Cartesian coordinates
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In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The adjective Cartesian refers to the French Mathematician and Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, the spherical and cylindrical coordinates for three-dimensional space. The development of the Cartesian coordinate system would play a fundamental role in the development of the Calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. A line with a chosen Cartesian system is called a number line.
Cartesian coordinates
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The right hand rule.
Cartesian coordinates
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinates
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3D Cartesian Coordinate Handedness
132.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, certain other spaces. It is named after the Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. Geometric shapes are defined as equations and inequalities. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. Euclidean spaces have finite dimension. One way to think of the Euclidean plane is as a set of points satisfying expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted by the same distance. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, so on. The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner.
Euclidean space
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A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Euclidean spaces
133.
Polar coordinates
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The reference point is called the pole, the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the angle is called the angular coordinate, polar angle, or azimuth. The concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. From the 8th onward, astronomers developed methods for calculating the direction from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis.
Polar coordinates
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Hipparchus
Polar coordinates
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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Polar coordinates
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A planimeter, which mechanically computes polar integrals
134.
Curvilinear coordinates
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In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation, locally invertible at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The coordinates, coined by the French Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of coordinate systems in Euclidean space are Cartesian, spherical polar coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates are often used to define the distribution of physical quantities which may be, for example, tensors. Such expressions then become valid for any curvilinear coordinate system. Depending on the application, a system may be simpler to use than the coordinate system. For instance, a physical problem with spherical symmetry defined in R3 is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as engineering. For now, consider 3d space.
Curvilinear coordinates
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Curvilinear, affine, and Cartesian coordinates in two-dimensional space
135.
Legendre polynomials
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In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation in spherical coordinates. The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. These solutions for n = 0, 1, 2... form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn is an nth-degree polynomial. It may be expressed using Rodrigues' formula: P n = 1 2 n n! D n d x n. The Pn can also be defined as the coefficients in a Taylor series expansion: In physics, this ordinary generating function is the basis for multipole expansions. Expanding the Taylor series in Equation for the first two terms gives P 0 = 1, P 1 = x for the first two Legendre Polynomials. This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively. In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram–Schmidt process on the polynomials with respect to this inner product. The series converges when r > r ′. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge.
Legendre polynomials
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Figure 2
136.
List of trigonometric identities
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Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. This article uses Greek letters such as alpha, theta to represent angles. Different units of measure are widely used, including degrees, radians, gradians: 1 full circle = 360 degrees = 2π radians = 400 gons. The following table shows the conversions and values for some common angles: Results for other angles can be found at Trigonometric constants expressed in real radicals. Angles ending in a symbol are in degrees. The trigonometric functions are the cosine of an angle. The tangent of an angle is the ratio of the sine to the cosine: θ = sin θ θ. These definitions are sometimes referred to as ratio identities. The trigonometric functions are inverse functions for the trigonometric functions. This follows for the unit circle. Where the sign depends on the quadrant of θ. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
List of trigonometric identities
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Cosines and sines around the unit circle
137.
Pythagorean trigonometric identity
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The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is given by the formula: 2 θ + cos θ = 1.. . This relation between cosine is sometimes called the fundamental trigonometric identity. Therefore, this trigonometric identity follows from the Pythagorean theorem. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the namely θ. Alternatively, the identities found at Trigonometric symmetry, shifts, periodicity may be employed. By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Tan θ = b a, and: sec θ = c a. In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem. The following table gives the identities with the divisor that relates them to the main identity. The circle centered at the origin in the Euclidean plane is defined by the equation: x 2 + 2 = 1. Consequently, from the equation for the unit circle: cos 2 θ + sin 2 θ = 1, the Pythagorean identity.
Pythagorean trigonometric identity
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Similar right triangles showing sine and cosine of angle θ
138.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. The ancient Nubians used a similar method. In the 2nd AD, the Greco-Egyptian Ptolemy printed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata. These Indian works were expanded by Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry.
Trigonometry
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Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".
Trigonometry
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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
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Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
139.
Cross product
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It has many applications in mathematics, physics, engineering, programming. It should not be confused with product. If two vectors have the same direction or if either one has zero length, then their cross product is zero. The cross product is distributive over addition. But if the product is limited with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, then only the cross product qualifies. The cross product of two vectors b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes a ∧ b is used, though this is avoided in mathematics to avoid confusion with the exterior product. If b are parallel, by the above formula, the cross product of a and b is the zero vector 0. Then, the n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i.e. b × a = −. Using the cross product requires the handedness of the coordinate system to be taken into account. If a left-handed system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction. This, however, creates a problem because transforming to another should not change the direction of n. The problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector.
Cross product
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The cross-product in respect to a right-handed coordinate system
140.
Dot product
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In mathematics, the dot product or scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the cosine of the angle between them. In three-dimensional space, the product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions. The product may be defined algebraically or geometrically. The geometric definition is based on the notions of distance. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In such a presentation, the notions of length and angles are not primitive. For instance, in three-dimensional space, the product of vectors and is: ⋅ = + + = 4 − 6 + 5 = 3. In Euclidean space, a Euclidean vector is a geometrical object that possesses both a direction. A vector can be pictured as an arrow. Its direction is the direction that the arrow points. The magnitude of a vector a is denoted by ∥ a ∥.
Dot product
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Scalar projection
141.
Seven-dimensional cross product
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In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R7 a vector a × b also in R7. Like the cross product in three dimensions, a × b is orthogonal both to a and to b. The seven-dimensional product has the same relationship to the octonions as the three-dimensional product does to the quaternions. In other dimensions there are binary products with bivector results. The product can be given by a table, such as the one here. This table, due to Cayley, gives ei and ej for each i, j from 1 to 7. This can be repeated for the other six components. There are one for each of the products satisfying the definition. The top left 3 × 3 corner of this table gives the cross product in three dimensions. The first property states that the product is perpendicular to its arguments, while the second property gives the magnitude of the product. A third statement of the condition is | x × y | = | x | | y | if = 0. Given the properties of bilinearity, magnitude, a nonzero cross product exists only in three and seven dimensions. In the three-dimensional cross product, unique, there are many possible binary cross products in seven dimensions. It is not unique.
Seven-dimensional cross product
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Fano planes for the two multiplication tables used here.
142.
Similar figures
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained with additional reflection. This means that either object can be reflected, as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure. It can be shown that two triangles having congruent angles are similar, the corresponding sides can be proved to be proportional. This is known as the AAA 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, 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. 2. All the corresponding sides have lengths in the same ratio: AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other.
Similar figures
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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.)
Similar figures
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Figures shown in the same color are similar
143.
Hippocrates of Chios
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Hippocrates of Chios was an ancient Greek mathematician, geometer, astronomer, who lived c. 470 – c. 410 BCE. He was born on the isle of Chios, where he originally was a merchant. After some misadventures he went to Athens, possibly for litigation. There he grew into a leading mathematician. On Chios, Hippocrates may have been a pupil of the astronomer Oenopides of Chios. The reductio ad argument has been traced to him. Only a famous, fragment of Hippocrates' Elements is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some Hippocratic lunes -- see Lune of Hippocrates. The strategy apparently was to divide a circle into a number of crescent-shaped parts. If it were possible to calculate the area of each of those parts, then the area of the circle as a whole would be known too. Only much later was it proven that this approach had no chance of success, because the factor pi is transcendental. In the century after Hippocrates at least four other mathematicians wrote their own Elements, steadily improving logical structure. In this way Hippocrates' pioneering work laid the foundation for Euclid's Elements, to remain the standard textbook for many centuries. Two other contributions by Hippocrates in the field of mathematics are noteworthy. He found a way to tackle the problem of ` duplication of the cube', the problem of how to construct a root.
Hippocrates of Chios
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The Lune of Hippocrates. Partial solution of the " Squaring the circle " task, suggested by Hippocrates. The area of the shaded figure is equal to the area of the triangle ABC. This is not a complete solution of the task (the complete solution is proven to be impossible with compass and straightedge).
144.
Pappus of Alexandria
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Nothing is known of his life, except that he had a son named Hermodorus, was a teacher in Alexandria. Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics, including recreational mathematics, doubling the polyhedra. Pappus flourished in the 4th century AD. In a period of general stagnation in mathematical studies, he stands out as a remarkable exception. "In this respect the fate of Pappus strikingly resembles that of Diophantus." The Suda states that Pappus was of the same age as Theon of Alexandria, who flourished in the reign of Emperor Theodosius I. This works out as October 18, 320 AD, so Pappus must have flourished c. 320 AD. The Suda enumerates other works of Pappus: Χωρογραφια οἰκουμενική, commentary on the 4 books of Ptolemy's Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Ὀνειροκριτικά. Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclid's Elements, on Ptolemy's Ἁρμονικά. These discoveries form, in fact, a text upon which Pappus enlarges discursively. The portions of Collection which has survived can be summarized as follows. We can only conjecture that the lost Book I, like Book II, was concerned with arithmetic, Book III being clearly introduced as beginning a new subject. The whole of Book II discusses a method of multiplication from an unnamed book by Apollonius of Perga.
Pappus of Alexandria
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Title page of Pappus's Mathematicae Collectiones, translated into Latin by Federico Commandino (1589).
Pappus of Alexandria
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Mathematicae collectiones, 1660
145.
De Gua's theorem
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De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves. Let U be a measurable subset of a k-dimensional subspace of R n. The subsets I of with exactly 2 elements are, and. However the theorem had also been known much earlier to René Descartes. Weisstein, Eric W. "de Gua's theorem". MathWorld. Sergio A. Alvarez: Note on an n-dimensional Pythagorean theorem, Carnegie Mellon University. De Gua's Theorem, Pythagorean theorem in 3-D — Graphical illustration and related properties of the tetrahedron. Kheyfits, Alexander. "The Theorem of Cosines for Pyramids". The College Mathematics Journal. Mathematical Association of America. 35: 385–388. JSTOR 4146849. Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.
De Gua's theorem
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tetrahedron with a right-angle corner in O
146.
Tetrahedron
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In geometry, a tetrahedron is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity. In a regular tetrahedron, not only are all its faces the same shape but so are all its edges. If alternated with regular octahedra they form the cubic honeycomb, a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two dual tetrahedra form a stellated octahedron or octangula. This form has Schläfli h.
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
147.
Cube (geometry)
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids. It has 12 edges, 8 vertices. The cube is also a square parallelepiped, a right rhombohedron. It is a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has octahedral symmetry. The cube has four orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The third correspond to the A2 and B2 Coxeter planes. The cube can also be projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a cube's surface with center and length of 2a is the locus of all points such that max = a. A cube has the largest volume among cuboids with a given area. Also, a cube has the largest volume among cuboids with the same linear size.
Cube (geometry)
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(Click here for rotating model)
Cube (geometry)
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These familiar six-sided dice are cube-shaped.
148.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. A Hilbert space is an abstract space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise frequently in mathematics and physics, typically as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, -- ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Geometric intuition plays an important role in many aspects of Hilbert theory. Exact analogs of the Pythagorean theorem and law hold in a Hilbert space. At a deeper level, projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. The latter space is often in the older literature referred to as the Hilbert space. The product takes two vectors x and y, produces a real number x · y. The product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is positive definite: for all vectors x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the product, satisfies these three properties is known as a inner product.
Hilbert space
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David Hilbert
Hilbert space
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The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.
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Inner product space
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of the angle between two vectors. They also provide the means of defining orthogonality between vectors. Inner product spaces are studied in functional analysis. An inner product naturally induces an associated norm, thus an inner space is also a normed vector space. A complete space with an inner product is called a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of complex numbers C. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write x, y ⟩ as ⟨ y | x ⟩, respectively y † x. This reverse order is now occasionally followed in the more abstract literature, taking ⟨ y ⟩ to be conjugate linear in x rather than y. There are technical reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain therefore has to have characteristic equal to 0. This immediately excludes finite fields.
Inner product space
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Geometric interpretation of the angle between two vectors defined using an inner product
150.
Euclidean spaces
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, certain other spaces. It is named after the Greek Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Greek geometry defined Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. Geometric shapes are defined as inequalities. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space of the same dimension. Euclidean spaces have finite dimension. One way to think of the Euclidean plane is as a set of points satisfying expressible in terms of angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, so on. The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner.
Euclidean spaces
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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
151.
Function (mathematics)
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An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f. In this example, if the input is 3, we may write f = 9. Likewise, if the input is 3, then the output is also 9, we may write f = 9. The input variable are sometimes referred to as the argument of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse as a solution of a equation. In the example above, f = x2, we have the ordered pair. More commonly the word "range" is used to mean, specifically the set of outputs. The image of this function is the set of non-negative real numbers. In analogy with arithmetic, it is possible to define addition, division of functions, in those cases where the output is a number.
Function (mathematics)
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A function f takes an input x, and returns a single output f (x). One metaphor describes the function as a "machine" or " black box " that for each input returns a corresponding output.
152.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. There are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a space. In the same vein, but in a more geometric sense, vectors representing displacements in three-dimensional space also form vector spaces. Infinite-dimensional vector spaces arise naturally as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of continuity. Among these topologies, those that are defined by inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Vector spaces are applied throughout mathematics, science and engineering. Furthermore, vector spaces furnish an coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading in geometry and abstract algebra. This is used in physics to describe velocities. Given any two such arrows, w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.
Vector space
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Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
153.
Euclidean vector
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In mathematics, physics, engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added according to vector algebra. A vector is what is needed to "carry" the A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating rotation around the Sun. The direction refers to the direction of displacement from A to B. Associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Many physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors. The concept of vector, as we know today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of line segments of the same orientation. Essentially he thus erected the first space of vectors in the plane.
Euclidean vector
154.
Parallelogram equality
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In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. Using the notation in the diagram on the right, the sides are. The general formula simplifies to the parallelogram law. In an inner space, the norm is determined using the inner product: ∥ x ∥ 2 = ⟨ x, x ⟩. All normed vector spaces have norms. Given a norm, one can evaluate both sides of the law above. A remarkable fact is that if the law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p-norm if and only if p = 2, the Euclidean norm or standard norm. For any norm satisfying the law, the inner product generating the norm is unique as a consequence of the polarization identity. Commutative property Inner product space Normed vector space Polarization identity Weisstein, Eric W. "Parallelogram Law". MathWorld.
Parallelogram equality
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A parallelogram. The sides are shown in blue and the diagonals in red.
155.
Orthogonality
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, engineering. The word comes from the Greek ὀρθός, meaning "upright", γωνία, meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y. The largest subspace of V, orthogonal to a given subspace is its orthogonal complement. Two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
Orthogonality
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The line segments AB and CD are orthogonal to each other.
156.
Cartesian coordinate system
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In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The adjective Cartesian refers to the French Mathematician and Philosopher René Descartes who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, the spherical and cylindrical coordinates for three-dimensional space. The development of the Cartesian coordinate system would play a fundamental role in the development of the Calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces. A line with a chosen Cartesian system is called a number line.
Cartesian coordinate system
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The right hand rule.
Cartesian coordinate system
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
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3D Cartesian Coordinate Handedness
157.
Spherical law of cosines
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. If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable. It can be obtained to the given one. A proof of the law of cosines can be constructed as follows. Let u, w denote the unit vectors from the center of the sphere to those corners of the triangle. Similarly, t b = w − u cos b sin b. We then have two plane triangles with a side in common: the triangle containing u, y and z and the one containing O, y and z. With this rotation, the spherical coordinates for v are = and the spherical coordinates for w are =. The Cartesian coordinates for w are =. Half-side formula Hyperbolic law of cosines Solution of triangles
Spherical law of cosines
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Spherical triangle solved by the law of cosines.
158.
Maclaurin series
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The concept of a Taylor series was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation. The polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function, equal to its Taylor series in an open interval is known as an analytic function in that interval. + f ″ 2! 2 + f ‴ 3! 3 + ⋯. Which can be written in the more compact sigma notation as ∑ n = 0 ∞ f n! N where n! Denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0! are both defined to be 1. When a = 0, the series is also called a Maclaurin series.
Maclaurin series
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As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
159.
Big O notation
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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, others, collectively called Bachmann–Landau notation or asymptotic notation. A famous example is the problem of estimating the term in the prime number theorem. Big O notation characterizes functions according to their growth rates: different functions with the same rate may be represented using the same O notation. The O is used because the growth rate of a function is also referred to as order of the function. A description of a function in terms of big O notation only provides an upper bound on the growth rate of the function. Big O notation is also used in other fields to provide similar estimates. G be two functions defined on some subset of the real numbers. If f is a product of several factors, any constants can be omitted. This function is the sum of three terms: 5. Of these three terms, the one with the highest rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f is a "big-oh" of. Mathematically, we can write f = O.
Big O notation
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Example of Big O notation: f (x) ∈ O(g (x)) as there exists c > 0 (e.g., c = 1) and x 0 (e.g., x 0 = 5) such that f (x) < c g (x) whenever x > x 0.
160.
Loss of significance
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Loss of significance is an undesirable effect in calculations using finite-precision arithmetic. It occurs when an operation on two numbers increases relative error substantially more than it increases absolute error, for example in subtracting two nearly equal numbers. The effect is that the number of significant digits in the result is reduced unacceptably. Ways to avoid this effect are studied in numerical analysis. The effect can be demonstrated with decimal numbers. It is very different when measured in order of precision. The first is accurate to 6981099999999999999 ♠ 10 10 − 20, while the second is only accurate to 6991100000000000000 ♠ 10 × 10 − 10. In the second case, the answer seems to have one significant digit, which would amount to loss of significance. Furthermore, it usually only postpones the problem: What if the data is accurate to only ten digits? The same effect will occur. One of the most important parts of numerical analysis is to minimize loss of significance in calculations. If the underlying problem is well-posed, there should be a stable algorithm for solving it. Let y be positive normalized floating point numbers. This formula may not always produce an accurate result. For example, when c is very small, loss of significance can occur in either of the root calculations, depending on the sign of b.
Loss of significance
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Example of LOS in case of computing 2 forms of the same function
161.
Hyperbolic triangle
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In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called three points called angles or vertices. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces. A hyperbolic triangle consists of the three segments between them. There is an upper bound for the area of triangles. There is an upper bound for radius of the inscribed circle. Two triangles are only if they correspond under a finite product of line reflections. Two triangles with corresponding angles equal are congruent. The area of a triangle is proportional to the deficit of its sum from 180 °. Hyperbolic triangles are thin, there is a maximum distance δ of the other two edges. This principle gave rise to δ-hyperbolic space. The definition of a triangle can be generalized, permitting vertices while keeping the sides within the plane. If a pair of sides is limiting parallel, then they end at an ideal vertex represented as an point. Such a pair of sides may also be said to form an angle of zero.
Hyperbolic triangle
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A hyperbolic triangle embedded in a saddle-shaped surface
162.
Gaussian curvature
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For example, a flat plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the inside of a torus. This is the content of the Theorema egregium. Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827. The Gaussian curvature is the product of the two principal curvatures Κ = κ1 κ2. The sign of the Gaussian curvature can be used to characterise the surface. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign. At such points, the surface will be saddle shaped. The surface is said to have a parabolic point. When a surface has a constant zero Gaussian curvature, the geometry of the surface is Euclidean geometry. When a surface has a constant positive Gaussian curvature, the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, the geometry of the surface is hyperbolic geometry. In geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends at that point.
Gaussian curvature
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From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).
163.
Hyperbolic function
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The hyperbolic functions are hyperbolic sine "arsinh" and on. Just as the points form a circle with a unit radius, the points form the right half of the equilateral hyperbola. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Laplace's equations are important in many areas including electromagnetic theory, special relativity. In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, are hence holomorphic. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. The abbreviations sh and ch are still used in some other languages, like French and Russian. The complex forms in the definitions above derive from Euler's formula. Sinh = sinh 2 = cosh − 2 where sgn is the function.
Hyperbolic function
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Hyperbolic functions in the complex plane
Hyperbolic function
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A ray through the unit hyperbola in the point, where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
Hyperbolic function
Hyperbolic function
164.
Plimpton 322
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Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This table lists what are now called Pythagorean triples, c satisfying a2 + b2 = c2. Although the tablet was interpreted in the past as a trigonometric table, more recently published work gives it a different function. For readable popular treatments of this tablet see Robson or, briefly Conway & Guy. Robson is a more technical discussion of the interpretation of the tablet's numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide, 2 cm thick. According to Banks, the tablet came to the ancient city of Larsa. Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table in Babylonian sexagesimal notation. The fourth column is just a row number, from 1 to 15. The third columns are completely visible in the surviving tablet. The sixty sexigesimal entries are no truncations or rounding off. Scholars still differ, however, on how these numbers were generated.
Plimpton 322
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The Plimpton 322 tablet.
165.
First Babylonian Dynasty
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The chronology of the first dynasty of Babylonia is debated as there is a Babylonian King List A and a Babylonian King List B. In this chronology, the regnal years of List A are used due to their wide usage. The reigns in List B are longer, in general. Thus any evidence must come from written records. Not much is known through Sin-muballit other than the fact they were Amorites rather than indigenous Akkadians. What is known, however, is that they accumulated little land. When Hammurabi ascended the throne of Babylon, the empire only consisted of a few towns in the surrounding area: Borsippa. Once Hammurabi was king, his military victories gained land for the empire. However, Babylon remained Larsa, then ruled by Rim-Sin I. In Hammurabi's thirtieth year as king, he really began to establish Babylon as the center of what would be a great empire. In that year, he conquered Larsa from Rim-Sin I, thus gaining control over the urban centers of Nippur, Ur, Uruk, Isin. In essence, Hammurabi gained control over all of south Mesopotamia. The other political power in the region in the 2nd millennium was Eshnunna, which Hammurabi succeeded in capturing in c. 1761. 1761. Babylon exploited the economic stability that came with them.
First Babylonian Dynasty
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The extent of the Paleo-Babylonian Empire at the start and end of Hammurabi of Babylon 's reign, c. 1792 BCE — c. 1750 BCE
166.
Stigler's law of eponymy
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Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy". It states that no scientific discovery is named after its original discoverer. Stigler himself named the sociologist Robert K. Merton as the discoverer of "Stigler's law" to show that it follows its own decree. Often, several people will arrive at a new idea around the same time, as in the case of calculus. It can be dependent on the publicity of the new work and the fame of its publisher as to whether the scientist's name becomes historically associated. He added his little mite —, all he did. But nothing can do that.” Stephen Stigler's father, the economist George Stigler, also examined the process of discovery in economics. He gave several examples in which the original discoverer was not recognized as such. Boyer's Law was named by Hubert Kennedy in 1972. Kennedy observed that "it is perhaps interesting to note that this is probably a rare instance of a law whose statement confirms its own validity". "Everything of importance has been said before by somebody who did not discover it" is an adage attributed to Alfred North Whitehead. The Economist as Preacher, Other Essays. Chicago: The University of Chicago Press. ISBN 0-226-77430-9.
Stigler's law of eponymy
167.
Yale
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Yale University is an American private Ivy League research university in New Haven, Connecticut. Founded in 1701 in Saybrook Colony to train Congregationalist ministers, it is the third-oldest institution of higher education in the United States. Originally restricted to theology and sacred languages, the curriculum began to incorporate sciences by the time of the American Revolution. Yale is organized into fourteen constituent schools: the original undergraduate college, twelve professional schools. While the university is governed by the Yale Corporation, each school's faculty oversees its degree programs. The university's assets include an endowment valued at the second largest of any educational institution. The Yale University Library, serving all constituent schools, is the third-largest academic library in the United States. Yale College undergraduates are organized into a social system of residential colleges. Almost all faculty teach undergraduate courses, more than 2,000 of which are offered annually. Students compete intercollegiately in the NCAA Division I Ivy League. Yale has graduated notable alumni, including five U.S. Presidents, 19 U.S. Supreme Court Justices, 13 living billionaires, many heads of state. In addition, Yale has graduated hundreds of members of many high-level U.S. diplomats. 52 Nobel laureates, 5 Fields Medalists, 119 Marshall Scholars have been affiliated with the University. The Act was an effort to create an institution to lay leadership for Connecticut. The group, led by James Pierpont, is now known as "The Founders".
Yale
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Charter creating Collegiate School, which became Yale College, October 9, 1701
Yale
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Yale University Seal
Yale
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A Front View of Yale-College and the College Chapel, Daniel Bowen, 1786.
Yale
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First diploma awarded by Yale College, granted to Nathaniel Chauncey, 1702.
168.
Leon Lederman
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He is Director Emeritus of Fermi National Accelerator Laboratory in Illinois, USA. He has been Resident Scholar Emeritus since 2012. In 2012, he was awarded the Vannevar Bush Award for his extraordinary contributions to understanding the basic particles of nature. Lederman was born in New York City, New York, Morris Lederman, a laundryman. Lederman graduated in the South Bronx. He received a Ph.D. from Columbia University in 1951. He then eventually became Eugene Higgins Professor of Physics. On leave from Columbia, he spent some time at CERN in Geneva as a Ford Foundation Fellow. He took an extended leave of absence from Columbia in 1979 to become director of Fermilab. In 1991, Lederman became President of the American Association for the Advancement of Science. Lederman is also one of the main proponents of the "Physics First" movement. A former president of Lederman also received the National Medal of Science, the Wolf Prize and the Ernest O. Lawrence Medal. Lederman served as President of the Board of Sponsors of The Bulletin of the Atomic Scientists. Among his achievements are the discovery of the muon neutrino in 1962 and the bottom quark in 1977. These helped establish his reputation as among the top particle physicists.
Leon Lederman
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Lederman on May 11, 2007
169.
Pythagorean triples
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A Pythagorean triple consists of three positive integers a, b, c, such that a2 + b2 = c2. A well-known example is. If is a Pythagorean triple, then so is for any positive k. A primitive Pythagorean triple is one in which a, c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples. Moreover, 1 and 2 do not have an integer common multiple because √ 2 is irrational. There are 16 primitive Pythagorean triples with ≤ 100: Note, for example, not a primitive Pythagorean triple, as it is a multiple of. Each of these low-c points forms one of the more easily recognizable radiating lines in the plot. The triple generated by Euclid's formula is only if m and n are coprime and not both odd. Every primitive triple arises from a unique pair of coprime numbers n, one of, even. It follows that there are infinitely primitive Pythagorean triples. This relationship of b and c to m and n from Euclid's formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples -- for example, can not be generated using integer n. This can be remedied by inserting an additional k to the formula.
Pythagorean triples
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The Pythagorean theorem: a 2 + b 2 = c 2
170.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. The letter c is a constant, the speed of light in a vacuum. Algebra gives methods for expressing formulas that are much easier than the older method of writing everything out in words. The algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. The algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa ` l-muḳābala by al-Khwarizmi. The word entered the English language from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting dislocated bones. The mathematical meaning was first recorded in the sixteenth century. The word "algebra" has related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.
Algebra
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A page from Al-Khwārizmī 's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
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Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
171.
Middle Kingdom of Egypt
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Some scholars also include the Thirteenth Dynasty of Egypt wholly into this period as well, in which case the Middle Kingdom would finish c. 1650, while others only include it until Merneferre Ay c. 1700 BC, last king of this dynasty to be attested in both Upper and Lower Egypt. During the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards, centered on el-Lisht. After the collapse of the Old Kingdom, Egypt entered a period of decentralization called the First Intermediate Period. Towards the end of this period, two rival dynasties, known as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt to the Tenth Nome of Upper Egypt. To the north, Lower Egypt was ruled by the 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B.C. During Mentuhotep II's fourteenth he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. For this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south far as the Second Cataract in Nubia, which had gained its independence during the First Intermediate Period. He also restored Egyptian hegemony over the Sinai region, lost since the end of the Old Kingdom. Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all Egyptian king lists. The Turin Papyrus claims after Mentuhotep III came "seven kingless years."
Middle Kingdom of Egypt
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An Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
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The head of a statue of Senusret I.
Middle Kingdom of Egypt
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Statue head of Senusret III
172.
Egypt
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It is the world's only contiguous Afrasian nation. Egypt has among the longest histories of emerging as one of the world's first nation states in the tenth millennium BC. Considered a cradle of civilisation, Ancient Egypt experienced some of the earliest developments of writing, agriculture, urbanisation, organised central government. One of the earliest centres of Christianity, Egypt remains a predominantly Muslim country, albeit with a significant Christian minority. The large regions of the Sahara desert, which constitute most of Egypt's territory, are sparsely inhabited. Egypt is a member of the United Nations, Non-Aligned Movement, Arab League, Organisation of Islamic Cooperation. Miṣr is modern official name of Egypt, while Maṣr is the local pronunciation in Egyptian Arabic. The name is of Semitic origin, directly cognate with Semitic words for Egypt such as the Hebrew מִצְרַיִם. The oldest attestation of this name for Egypt is the Akkadian KURmi-iṣ-ru miṣru, meaning "border" or "frontier". There is evidence of rock carvings in desert oases. In the 10th millennium BC, a culture of fishers was replaced by a grain-grinding culture. Climate changes or overgrazing around 8000 BC began forming the Sahara. Early tribal peoples migrated to the Nile River where they developed more centralised society. By about 6000 BC, a Neolithic culture rooted in the Nile Valley. During the Neolithic era, predynastic cultures developed independently in Upper and Lower Egypt.
Egypt
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The Giza Necropolis is the oldest of the ancient Wonders and the only one still in existence.
Egypt
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Flag
Egypt
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The Greek Ptolemaic queen Cleopatra VII and her son by Julius Caesar, Caesarion at the Temple of Dendera.
Egypt
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The 1803 Cedid Atlas, showing Ottoman Egypt.
173.
Berlin Papyrus 6619
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The two readable fragments were published by Hans Schack-Schackenburg in 1902. The papyrus is one of the primary sources of Egyptian mathematics. The Berlin Papyrus contains two problems, the first stated as "the area of a square of 100 is equal to that of two smaller squares. The side of one is + 1/4 the side of the other." Papyrology Timeline of mathematics Egyptian fraction Simultaneous equation examples from the Berlin papyrus Two algebra problems compared to RMP algebra Two suggested solutions
Berlin Papyrus 6619
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Berlin Papyrus 6619, as reproduced in 1900 by Schack-Schackenburg
174.
Hammurabi
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Hammurabi was the sixth king of the First Babylonian Dynasty, reigning from 1792 BC to 1750 BC. He was preceded by his father, Sin-Muballit, who abdicated due to failing health. He extended Babylon's control throughout Mesopotamia through military campaigns. Hammurabi is known for the Code of Hammurabi, one of the earliest surviving codes of law in recorded history. The name Hammurabi derives from the Amorite term ʻAmmurāpi, itself from ʻAmmu and Rāpi. Hammurabi was an Amorite First Dynasty king of the city-state of Babylon, inherited the power from his father, Sin-Muballit, in c. 1792 BC. Though many cultures co-existed in Mesopotamia, Babylonian culture gained a degree of prominence among the literate classes throughout the Middle East under Hammurabi. The kings who came before Hammurabi had founded a relatively minor City State in 1894 BC which controlled little territory outside of the city itself. Babylon was overshadowed by older, larger and more powerful kingdoms such as Elam, Assyria, Isin, Eshnunna and Larsa for a century or so after its founding. Thus Hammurabi ascended to the throne as the king of a minor kingdom in the midst of a complex geopolitical situation. The powerful kingdom of Eshnunna controlled the upper Tigris River while Larsa controlled the river delta. To the east of Mesopotamia lay the powerful kingdom of Elam which regularly invaded and forced tribute upon the small states of southern Mesopotamia. The first few decades of Hammurabi's reign were quite peaceful. Hammurabi used his power to undertake a series of public works, including heightening the city walls for defensive purposes, expanding the temples. In c. 1801 BC, the powerful kingdom of Elam, which straddled important trade routes across the Zagros Mountains, invaded the Mesopotamian plain.
Hammurabi
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Hammurabi (standing), depicted as receiving his royal insignia from Shamash (or possibly Marduk). Hammurabi holds his hands over his mouth as a sign of prayer (relief on the upper part of the stele of Hammurabi's code of laws).
Hammurabi
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This bust, known as the "Head of Hammurabi", is now thought to predate Hammurabi by a few hundred years (Louvre)
Hammurabi
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Code of Hammurabi stele. Louvre Museum, Paris
Hammurabi
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The bas-relief of Hammurabi at the United States Congress
175.
India
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India, officially the Republic of India, is a country in South Asia. It is the most populous democracy in the world. It is bounded on the southeast. It shares land borders with Pakistan to the west; China, Nepal, Bhutan to the northeast; and Myanmar and Bangladesh to the east. In the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. India's Andaman and Nicobar Islands share a maritime border with Thailand and Indonesia. Its capital is New Delhi; other metropolises include Mumbai, Kolkata, Chennai, Bangalore, Hyderabad and Ahmedabad. In 2016, the Indian economy was the world's sixth-largest by nominal GDP and third-largest by purchasing parity. Following economic reforms in 1991, India became one of the fastest-growing major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, inadequate public healthcare. It has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system and consists of 29 states and 7 union territories. It is also home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu. The latter term stems from the Sanskrit Sindhu, the historical local appellation for the Indus River.
India
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Flag
India
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The granite tower of Brihadeeswarar Temple in Thanjavur was completed in 1010 CE by Raja Raja Chola I.
India
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Writing the will and testament of the Mughal king court in Persian, 1590–1595
176.
Isosceles
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles. In an isosceles triangle that has exactly two equal sides, the equal sides are called the third side is called the base. The base is called the apex. Whether the isosceles triangle is acute, obtuse depends on the vertex angle. The Euler line of any triangle goes through the triangle's orthocenter, its circumcenter. In an isosceles triangle with exactly two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows. If the angle is acute, then the orthocenter, the centroid, the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line. The Steiner inellipse of any triangle is the unique ellipse, internally tangent to the triangle's three sides at their midpoints. This is what Heron's formula reduces in the isosceles case. This is derived by drawing a line from the base of the triangle, which bisects the vertex angle and creates two right triangles. The bases of these two right triangles are both equal by definition of the term "sine". For the same reason, the heights of these triangles are equal to the hypotenuse times the cosine of the bisected angle.
Isosceles
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Isosceles triangle with vertical axis of symmetry
177.
Zhoubi Suanjing
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The Zhou Bi Suan Jing, or Chou Pei Suan Ching, is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient dynasty Zhou c. 1046-771 BCE; "Bi" means thigh and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical calculation. "Suan Jing" or "classic of arithmetic" were appended in later time to honor the achievement of the book in mathematics. Its compilation and addition of materials continued into the Han Dynasty. It is an anonymous collection of 246 problems encountered by his astronomer and mathematician, Shang Gao. Each question has stated their numerical answer and corresponding algorithm. This book contains one of the first recorded proofs of the Pythagorean Theorem. Commentators such as Liu Hui, Zu Geng, Yang Hui have expanded on this text. Tsinghua Bamboo Slips Boyer, Carl B. A History of Mathematics, John Wiley & Sons, Inc. 2nd edition. ISBN 0-471-54397-7. Full text of the Zhou Bi Suan Jing, including diagrams - Chinese Text Project. Full text of the Zhou Bi Suan Jing, at Project Gutenberg Christopher Cullen. Astronomy and Mathematics in Ancient China: The'Zhou Bi Suan Jing', Cambridge University Press, 2007.
Zhoubi Suanjing
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing
Zhoubi Suanjing
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History
178.
China
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China, officially the People's Republic of China, is a unitary sovereign state in East Asia. With a population of over billion, it is the world's most populous country. The state is governed based in the capital of Beijing. The country's urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin and Hong Kong. China has been characterized as a potential superpower. Mountain ranges separate China from much of South and Central Asia. The third and sixth longest in the world, respectively, run from the Tibetan Plateau to the densely populated eastern seaboard. China's coastline along the Pacific Ocean is bounded by the Bohai, Yellow, East China, South China seas. China emerged in the fertile basin of the Yellow River in the North China Plain. For millennia, China's political system was based on hereditary monarchies known as dynasties. Since 221 BC, when the Qin Dynasty first conquered several states to form a Chinese empire, the state has expanded, reformed numerous times. Since the introduction of economic reforms in 1978, China has become one of the world's fastest-growing major economies. As of 2014, it is largest by purchasing power parity. China is also second-largest importer of goods. China has the world's largest standing army and second-largest defense budget.
China
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Yinxu, ruins of an ancient palace dating from the Shang Dynasty (14th century BCE)
China
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Flag
China
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Some of the thousands of life-size Terracotta Warriors of the Qin Dynasty, c. 210 BCE
China
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The Great Wall of China was built by several dynasties over two thousand years to protect the sedentary agricultural regions of the Chinese interior from incursions by nomadic pastoralists of the northern steppes.
179.
Han Dynasty
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The Han dynasty was the second imperial dynasty of China, preceded by the Qin dynasty and succeeded by the Three Kingdoms period. Spanning over four centuries, the Han period is considered a golden age in Chinese history. To this day, China's majority ethnic group refers to itself as the Chinese script is referred to as "Han characters". This interregnum separates the Han dynasty into two periods: the Eastern Han or Later Han. The emperor was at the pinnacle of Han society. He shared power with both the nobility and appointed ministers who came largely from the scholarly gentry class. These kingdoms gradually lost all vestiges of their independence, particularly following the Rebellion of the Seven States. This policy endured in AD 1911. The Han dynasty saw a significant growth of the money economy first established during the Zhou dynasty. The coinage issued by the central mint in 119 BC remained the standard coinage of China until the Tang dynasty. The period saw a number of institutional innovations. Emperor Wu of Han launched military campaigns against them. The ultimate Han victory in these wars eventually forced the Xiongnu to accept vassal status as Han tributaries. The territories north of Han's borders were quickly overrun by the nomadic Xianbei confederation. Imperial authority was also seriously challenged by large religious societies which instigated the Yellow Turban Rebellion and the Five Pecks of Rice Rebellion.
Han Dynasty
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History of China
Han Dynasty
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Han dynasty in 1 AD.
Han Dynasty
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A silk banner from Mawangdui, Changsha, Hunan province. It was draped over the coffin of Lady Dai (d. 168 BC), wife of the Marquess Li Cang (利蒼) (d. 186 BC), chancellor for the Kingdom of Changsha.
Han Dynasty
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A gilded bronze oil lamp in the shape of a kneeling female servant, dated 2nd century BC, found in the tomb of Dou Wan, wife of the Han prince Liu Sheng; its sliding shutter allows for adjustments in the direction and brightness in light while it also traps smoke within the body.
180.
The Nine Chapters on the Mathematical Art
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This book is one of the earliest surviving mathematical texts from Zhou Bi Suan Jing. These were commented on by Liu Hui in the 3rd century. There is also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan. Its influence on mathematical thought in China persisted until the Qing Dynasty era. Liu Hui wrote a very detailed commentary on this book in 263. Liu's commentary is of great mathematical interest in its own right. The Nine Chapters is an anonymous work, its origins are not clear. This is no longer the case. Writings on reckoning is an Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of texts known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. Contents of The Nine Chapters are as follows: 方田 Fangtian - Rectangular fields.
The Nine Chapters on the Mathematical Art
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A page of The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical Art
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History
181.
Plutarch
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Plutarch was a Greek biographer and essayist, known primarily for his Parallel Lives and Moralia. He is classified as a Middle Platonist. Plutarch's surviving works were intended for both Greek and Roman readers. His family was wealthy. The name of Plutarch's grandfather was Lamprias, as he attested in his Life of Antony. Timon and Lamprias, are frequently mentioned in his essays and dialogues, wherein Timon in particular is spoken of in the most affectionate terms. Rualdus, in his 1624 Life of Plutarchus, recovered the name of Plutarch's wife, Timoxena, from internal evidence afforded by his writings. Interestingly, he hinted in that letter of consolation. The exact number of his sons is not certain, although Autobulus and second Plutarch, are often mentioned. This is nowhere definitely stated. Plutarch studied mathematics and philosophy at the Academy of Athens from 66 to 67. At some point, Plutarch took up Roman citizenship. He was initiated into the mysteries of the Greek god Apollo. At his estate, guests from all over the empire congregated for serious conversation, presided over by Plutarch in his marble chair. The 78 essays and other works which have survived are now known collectively as the Moralia.
Plutarch
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Ruins of the Temple of Apollo at Delphi, where Plutarch served as one of the priests responsible for interpreting the predictions of the oracle.
Plutarch
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Parallel Lives, Amyot translation, 1565
Plutarch
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Plutarch's bust at Chaeronea, his home town.
Plutarch
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A page from the 1470 Ulrich Han printing of Plutarch's Parallel Lives.
182.
Cicero
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Marcus Tullius Cicero was a Roman philosopher, politician, lawyer, orator, political theorist, consul, constitutionalist. Cicero is considered one of Rome's greatest orators and prose stylists. He created a Latin philosophical vocabulary distinguishing himself as a translator and philosopher. Though he was an accomplished successful lawyer, he believed his political career was his most important achievement. Following Julius Caesar's death, he became an enemy of Mark Antony in the ensuing struggle, attacking him in a series of speeches. His severed hands and head were then, as a final revenge of Mark Antony, displayed in the Roman Forum. Petrarch's rediscovery of Cicero's letters is often credited for initiating the 14th-century Renaissance in public affairs, classical Roman culture. He was born in 106 BC in a hill town 100 kilometers southeast of Rome. His father possessed good connections in Rome. However, being a semi-invalid, Cicero studied extensively to compensate. Cicero's brother Quintus wrote in a letter that she was a thrifty housewife. Personal surname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was originally given to one of Cicero's ancestors who had a cleft in the tip of his nose resembling a chickpea. However, it is more likely that Cicero's ancestors prospered through the sale of chickpeas. Romans often chose personal surnames: the famous family names of Fabius, Lentulus, Piso come from the Latin names of beans, lentils, peas.
Cicero
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A first century AD bust of Cicero in the Capitoline Museums, Rome
Cicero
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The Young Cicero Reading by Vincenzo Foppa (fresco, 1464), now at the Wallace Collection
Cicero
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Cicero Denounces Catiline, fresco by Cesare Maccari, 1882–88
Cicero
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Cicero's death (France, 15th century)
183.
Plato
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Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, especially the Western tradition. Unlike nearly all of his philosophical contemporaries, Plato's entire œuvre is believed to have survived intact for over 2,400 years. Along with his teacher, his most famous student, Aristotle, Plato laid the very foundations of Western philosophy and science. Alfred North Whitehead once noted: "the safest general characterization of the philosophical tradition is that it consists of a series of footnotes to Plato." Friedrich Nietzsche, amongst other scholars, called Christianity, "Platonism for the people." Plato was the innovator of dialectic forms in philosophy, which originate with him. ... ... He was not the first writer to whom the word "philosopher" should be applied. Due to a lack of surviving accounts, little is known about education. The philosopher came in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies. It is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born between 429 and 423 BCE. His father was Ariston.
Plato
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Plato: copy of portrait bust by Silanion
Plato
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Plato from The School of Athens by Raphael, 1509
Plato
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Plato and Socrates in a medieval depiction
Plato
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Plato (left) and Aristotle (right), a detail of The School of Athens, a fresco by Raphael. Aristotle gestures to the earth, representing his belief in knowledge through empirical observation and experience, while holding a copy of his Nicomachean Ethics in his hand. Plato holds his Timaeus and gestures to the heavens, representing his belief in The Forms
184.
Universum (UNAM)
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It was opened in 1992 at the Ciudad Universitaria in Mexico City. It has thirteen halls divided by theme dedicated to various permanent exhibitions. Opened in 1992, Universum is one of the first science museums of its type in Latin America. Its facilities cover 25,000 m2 with 12,000 m2 dedicated to permanent exhibitions. These permanent exhibitions are housed in main halls with various themes. Its mission is to contribute by society in general. For this reason, the exhibitions are designed to increase understanding by using attractive presentations. Another of the museum's functions is to make available work done by UNAM's researchers at the facility's library and archives. During the institution's twenty years, it has received over million visitors. Universum has also worked to create extension museums in other parts of the country such as the Museo de Ciencia y Tecnología in Chiapas. Permanent exhibits are housed on three floors of the main complex. A number of its exhibits have been developed with private and public entities. The Paráka Butterfly Exhibit is an area with live butterflies in an enclosure designed to imitate their natural habitat. It also works to breed butterflies native to the Valley of Mexico. The Health section focuses with interactive displays, some of which provide personal information.
Universum (UNAM)
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Main entrance to the museum
185.
John Aubrey
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John Aubrey FRS, was an English antiquary, natural philosopher and writer. He is perhaps best known as the author of his collection of short biographical pieces. He was also a folklorist, collecting together a miscellany of material on customs, traditions and beliefs under the title "Remaines of Gentilisme and Judaisme". He set out to compile county histories of both Wiltshire and Surrey, although both projects remained unfinished. His "Interpretation of Villare Anglicanum" was the first attempt to compile a full-length study of English place-names. He was friendly with many of the greatest scientists of the day. Only in the 1970s did the full breadth and innovation of his scholarship begin to be more widely appreciated. Many of his most important manuscripts remain unpublished, or published only in partial and unsatisfactory form. Aubrey was born at Easton Piers or Percy, to a long-established and affluent gentry family with roots in the Welsh Marches. Isaac Lyte, lived at Lytes Cary Manor, Somerset, now owned by the National Trust. His father, owned lands in Wiltshire and Herefordshire. For an only child, he was educated at home with a private tutor, he was "melancholy" in his solitude. His father was not intellectual, preferring field sports to learning. Aubrey studied geometry in secret. He was educated under Robert Latimer.
John Aubrey
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John Aubrey
John Aubrey
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Part of the southern inner ring at Avebury
John Aubrey
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An early photograph of Stonehenge taken July 1877
186.
Thomas Hobbes
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Thomas Hobbes, in some older texts Thomas Hobbes of Malmesbury, was an English philosopher, best known today for his work on political philosophy. His 1651 Leviathan established social contract theory, the foundation of most later Western political philosophy. He was one of the founders of political science. Thomas Hobbes was born on 5 April 1588. His mother's name is unknown. Thomas Sr. was the vicar of Charlton and Westport. The younger, had a brother Edmund, about two years older, a sister. Thomas Sr. was involved with the local clergy outside his church forcing him to leave London and abandon the family. The family was left with no family. Around 1603 he went up to Magdalen Hall, the predecessor college to Hertford College, Oxford. He had some influence on Hobbes. At university, Hobbes appears to have followed his own curriculum; he was "little attracted by the scholastic learning". They both took part in a grand tour of Europe in 1610. Hobbes was exposed to European scientific and critical methods in contrast to the scholastic philosophy which he had learned in Oxford. His Cavendish, then the Earl of Devonshire, died of the plague in June 1628.
Thomas Hobbes
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Thomas Hobbes
Thomas Hobbes
Thomas Hobbes
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Frontispiece from De Cive (1642)
Thomas Hobbes
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Frontispiece of Leviathan
187.
Hans Christian Andersen
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Hans Christian Andersen was a Danish author. Although a prolific writer of plays, travelogues, poems, Andersen is best remembered for his fairy tales. Andersen's popularity is not limited to children; his stories, called eventyr in Danish, express themes that transcend nationality. His stories plays. Hans Christian Andersen was born on 2 April 1805. He was an only child. Andersen's father, also Hans, considered himself related to nobility. Investigations prove these stories unfounded. This theory has been criticized. Andersen's father, who had received an elementary education, introduced Andersen to literature, reading to him Arabian Nights. Anne Marie Andersdatter, was uneducated and worked as a washerwoman following his father's death in 1816; she remarried in 1818. At 14, he moved to Copenhagen to seek employment as an actor. His voice soon changed. A colleague at the theatre told him that he considered Andersen a poet. Taking the suggestion seriously, Andersen began to focus on writing.
Hans Christian Andersen
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Photograph taken by Thora Hallager, 1869
Hans Christian Andersen
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Andersen's childhood home in Odense
Hans Christian Andersen
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Paper chimney sweep cut by Andersen
Hans Christian Andersen
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Painting of Andersen, 1836, by Christian Albrecht Jensen
188.
Major-General's Song
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"I Am the Very Model of a Modern Major-General" is a patter song from Gilbert and Sullivan's 1879 comic opera The Pirates of Penzance. It is perhaps the most famous song in Gilbert and Sullivan's operas. It is sung towards the end of Act I. The song satirises the idea of the "modern" educated British Army officer of the 19th century. It is one of the most difficult patter songs to perform, due to the fast pace and nature of the lyrics. The stage directions in the libretto state that at the end of each verse the Major-General is "bothered for a rhyme". In each case he finds a rhyme and finishes the verse with a flourish. The character of Major-General Stanley was widely taken to be a caricature of the general Sir Garnet Wolseley. Gilbert disliked Turner, who, unlike the progressive Wolseley, was of the old school of officers. Nevertheless, in George Grossmith imitated Wolseley's mannerisms and appearance, particularly his large moustache, the audience recognised the allusion. Notes: The Pirate Movie, a 1982 modern musical parody of The Pirates of Penzance, features many songs from the opera, including this song. I'm younger than The Rolling Stones." In the 1983 film Never Cry Wolf, the hero sings the song. Parts of it, has been sung in numerous television programs. For example, The Muppet Show staged a duet of the song with a 7-foot-tall talking carrot.
Major-General's Song
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Drawing from 1884 children's Pirates
Major-General's Song
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Henry Lytton as the Major-General (1919).
Major-General's Song
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1880 poster
189.
Gilbert and Sullivan
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The two men collaborated on fourteen comic operas between 1871 and 1896, of which H.M.S. Pinafore, The Pirates of Penzance and The Mikado are among the best known. Six years Gilbert's junior, composed the music, contributing memorable melodies that could convey both humour and pathos. Their operas are still performed frequently throughout the English-speaking world. Gilbert and Sullivan introduced innovations in form that directly influenced the development of musical theatre through the 20th century. The operas have been widely parodied and pastiched by humorists. Producer Richard D'Oyly Carte nurtured their collaboration. Gilbert was born in London on 18 November 1836. William, was a naval surgeon who later wrote novels and short stories, some of which included illustrations by his son. Playwright Mike Leigh described the "Gilbertian" style as follows: With great fluidity and freedom, continually challenges our natural expectations. First, within the framework of the story, he turns the world on its head. His genius is to fuse opposites with an imperceptible sleight of hand, to blend the surreal with the real, the caricature with the natural. In other words, to tell a perfectly outrageous story in a completely deadpan way. Gilbert developed his innovative theories following theatrical reformer Tom Robertson. At the time Gilbert began writing, theatre in Britain was in disrepute.
Gilbert and Sullivan
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W. S. Gilbert
Gilbert and Sullivan
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Arthur Sullivan
Gilbert and Sullivan
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One of Gilbert's illustrations for his Bab Ballad "Gentle Alice Brown"
Gilbert and Sullivan
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An early poster showing scenes from The Sorcerer, Pinafore, and Trial by Jury
190.
The Pirates of Penzance
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The story concerns Frederic, who, having completed his 21st year, is released to a band of tender-hearted pirates. The two young people fall instantly in love. So, technically, he has a birthday only once each leap year. His indenture specifies that he remain apprenticed to the pirates until his "twenty-first birthday", meaning that he must serve for another 63 years. Bound by his own sense of duty, Frederic's only solace is that Mabel agrees to wait for him faithfully. Pirates introduced the much-parodied "Major-General's Song". The opera was performed for over a century by many other opera companies and repertory companies worldwide. Pirates remains popular today, taking its place along with The Mikado and H.M.S. Pinafore as one of the most frequently played Gilbert and Sullivan operas. The Pirates of Penzance was the only Gilbert and Sullivan opera to have its official premiere in the United States. At the time, American law offered no protection to foreigners. After the pair's previous opera, H.M.S. Fiction and plays about pirates were ubiquitous in the 19th century. James Fenimore Cooper's The Red Rover were key sources for the romanticised, dashing pirate image and the idea of repentant pirates. Both Gilbert and Sullivan had parodied these ideas early in their careers.
The Pirates of Penzance
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Drawing of the Act I finale
The Pirates of Penzance
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The Pirate Publisher – An International Burlesque that has the Longest Run on Record, from Puck, 1886: Gilbert is seen as one of the British authors whose works are stolen by the pirate publisher.
The Pirates of Penzance
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George Grossmith as General Stanley, wearing Wolseley 's trademark moustache
The Pirates of Penzance
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Catherine Ferguson (Kate), Nellie Briercliffe (Edith), and Ella Milne (Isabel), 1920
191.
Scarecrow (Oz)
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The Scarecrow is a character in the fictional Land of Oz created by American author L. Frank Baum and illustrator W.W. Denslow. In his first appearance, the Scarecrow reveals that he lacks a brain and desires above all else to have one. In reality, he is merely ignorant. The old crow then told the Scarecrow of the importance of brains. The "mindless" Scarecrow joins Dorothy in the hope that The Wizard will give a brain. They are later joined by the Cowardly Lion. When the group goes to the West, he kills the Witch's crows by twisting their necks. After her friends have completed their mission to kill the Wicked Witch of the West, the Wizard gives the Scarecrow brains. Before he leaves Oz in a balloon, the Wizard appoints the Scarecrow to rule the Emerald City in his stead. Indeed, both believe they have neither. Symbolically, because they remain throughout her quest, she is provided with both and need not select. In the musical of Gregory Maguire's interpretation of the Oz franchise, this version of the Scarecrow was the love interest of Elphaba Thropp. Fiyero attends the Good, while they were all still young. Fiyero takes a special interest in Elphaba.
Scarecrow (Oz)
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Denslow's drawing of scarecrow hung up on pole and helpless, from first edition of book, 1900
Scarecrow (Oz)
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July 1896 Puck cartoon shows farmer hung up on pole and helpless; was this Denslow's inspiration? The hat says "silverite"; the locomotive is gold
Scarecrow (Oz)
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Cover of The Scarecrow of Oz (1915) by L. Frank Baum; illustration by John R. Neill
192.
The Wizard of Oz (1939 film)
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The film stars Judy Garland as Dorothy Gale. Notable for its use of Technicolor, unusual characters, over the years, it has become an icon of American popular culture. It was nominated for six Academy Awards, including Best Picture, but lost to Gone with the Wind. It did win in two other categories, including Best Original Song for "Over the Rainbow" and Best Original Score by Herbert Stothart. However, the film was a disappointment on its initial release, earning only $3,017,000 on a $2,777,000 budget, despite receiving largely positive reviews. Designation on the registry calls for efforts to preserve it for being "culturally, historically, aesthetically significant". It is also one of the few films on UNESCO's Memory of the World Register. The Wizard of Oz is often ranked on best-movie lists in critics' and public polls. It is the source of many quotes referenced in modern popular culture. It was directed primarily by Victor Fleming. Noel Langley, Florence Ryerson, Edgar Allan Woolf received credit for the screenplay, but uncredited contributions were made by others. The songs were written by Edgar "Yip" Harburg and Harold Arlen. The musical score and the incidental music were composed by Stothart. The film begins in Kansas, depicted in a sepia tone. Dorothy Gale lives with her dog Toto on the farm of her Aunt Em and Uncle Henry.
The Wizard of Oz (1939 film)
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Theatrical release poster
The Wizard of Oz (1939 film)
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Dorothy (Judy Garland, right) with Glinda the Good Witch of the North (Billie Burke)
The Wizard of Oz (1939 film)
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The film's main characters (left to right): the Cowardly Lion, Dorothy, Scarecrow, and the Tin Man
The Wizard of Oz (1939 film)
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Buddy Ebsen 's first makeup test as the Tin Man.
193.
Wizard (Oz)
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Oscar Zoroaster Phadrig Isaac Norman Henkle Emmannuel Ambroise Diggs is a fictional character in the Land of Oz created by American author L. Frank Baum. The character was further popularized by the classic 1939 movie, wherein his full name is not mentioned. The Wizard is one of the characters in The Wonderful Wizard of Oz. Unseen for most of the novel, he is the ruler of the Land of Oz and highly venerated by his subjects. Eventually each is granted one by one. When, at last, he grants an audience to all of them at once, he seems to be a disembodied voice. Working as a magician for a circus, he wrote OZ on the side of his hot air balloon for promotional purposes. One day his balloon sailed into the Land of Oz, he found himself worshipped as a great sorcerer. As Oz had no leadership at the time, he became Supreme Ruler of the kingdom, did his best to sustain the myth. He leaves Oz in a hot balloon. In The Marvelous Land of Oz, the Wizard is described as having handed to Mombi. The Wizard returns in the novel Dorothy and the Wizard in Oz. With the Zeb, he falls through a crack in the earth; in their underground journey, he acts as their guide and protector. Oz explains that his real name is Oscar Zoroaster Phadrig Isaac Norman Henkle Emmannuel Ambroise Diggs. Since they spell out the pinhead, he shortened his name further and called himself "Oz".
Wizard (Oz)
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Oscar Diggs aka the Wizard--illustration by William Wallace Denslow (1900)
194.
Uganda
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Uganda, officially the Republic of Uganda, is a landlocked country in East Africa. Uganda is the world's second most landlocked country after Ethiopia. The southern part of the country includes a substantial portion of Lake Victoria, shared with Kenya and Tanzania. Uganda is in the African Great Lakes region. Uganda also has a varied but generally a modified equatorial climate. Uganda takes its name from the Buganda kingdom, which encompasses a large portion including the capital Kampala. The people of Uganda were hunter-gatherers until 1,700 to 2,300 years ago, when Bantu-speaking populations migrated to the southern parts of the country. Beginning in 1894, the area was ruled by the British, who established administrative law across the territory. Uganda gained independence on 9 October 1962. The official language is English. Several other languages are also spoken including Runyoro, Runyankole, Rukiga, Luo. The president of Uganda is Yoweri Museveni, who came after a protracted six-year guerrilla war. The ancestors of the Ugandans were hunter-gatherers until 1,700-2,300 years ago. Bantu-speaking populations, who were probably from central Africa, migrated to the southern parts of the country. These groups developed ironworking skills and new ideas of social and political organisation.
Uganda
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Left: Construction of the Owen Falls Dam in Jinja. Right: A street in Uganda in the 1950s.
Uganda
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Flag
Uganda
Uganda
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Mount Kadam, Uganda.
195.
Greece
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Greece, officially the Hellenic Republic, historically also known as Hellas, is a country in southeastern Europe. Greece's population is approximately million as of 2015. Athens is largest city, followed by Thessaloniki. Greece is strategically located at the crossroads of Europe, Asia, Africa. Greece consists of nine geographic regions: Macedonia, Central Greece, the Peloponnese, Thessaly, Epirus, the Aegean Islands, Thrace, Crete, the Ionian Islands. The Aegean Sea lies to the south. Eighty percent of Greece is mountainous, with Mount Olympus being the highest peak at 2,918 metres. From the eighth BC, the Greeks were organised into various independent city-states, known as polis, which spanned the entire Mediterranean region and the Black Sea. The establishment of the Greek Orthodox Church in the first century transmitted Greek traditions to the wider Orthodox World. Falling under Ottoman dominion in the mid-15th century, the modern state of Greece emerged in 1830 following a war of independence. Greece's historical legacy is reflected by its 18 UNESCO World Heritage Sites, among the most in Europe and the world. Greece is a democratic and developed country with an advanced high-income economy, a very high standard of living. A founding member of the United Nations, Greece has been part of the Eurozone since 2001. Large tourism industry, prominent shipping sector and geostrategic importance classify it as a middle power. It is one of the most visited the largest economy in the Balkans, where it is an important regional investor.
Greece
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Fresco displaying the Minoan ritual of "bull leaping", found in Knossos, Crete.
Greece
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Flag
Greece
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The Lion Gate, Mycenae
Greece
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The Parthenon on the Acropolis of Athens is one of the best known symbols of classical Greece.
196.
Japan
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Japan is an island nation in East Asia. It is often called the "Land of the Rising Sun". Japan is a stratovolcanic archipelago of 6,852 islands. Largest are Honshu, Hokkaido, Kyushu and Shikoku, which make up about ninety-seven percent of Japan's land area. The country is divided into 47 prefectures in eight regions. The population of million is the world's tenth largest. Japanese people make up 98.5% of Japan's total population. Approximately million people live in the core city of Tokyo, the capital of Japan. Archaeological research indicates that Japan was inhabited early as the Upper Paleolithic period. The first written mention of Japan is in Chinese history texts from the 1st AD. Influence from other regions, mainly China, followed from Western Europe, has characterized Japan's history. From the 12th century until 1868, Japan was ruled by successive military shoguns who ruled in the name of the Emperor. Since adopting its revised constitution in 1947, Japan has maintained a constitutional monarchy with an Emperor and an elected legislature called the National Diet. Japan is considered a great power. The country has the world's fourth-largest economy by purchasing power parity.
Japan
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The Golden Hall and five-storey pagoda of Hōryū-ji, among the oldest wooden buildings in the world, National Treasures, and a UNESCO World Heritage Site
Japan
Japan
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Samurai warriors face Mongols, during the Mongol invasions of Japan. The Kamikaze, two storms, are said to have saved Japan from Mongol fleets.
Japan
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Samurai could kill a commoner for the slightest insult and were widely feared by the Japanese population. Edo period, 1798
197.
San Marino
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Its size is just over 61 km2, with a population of 33,562. Its largest city is Dogana. San Marino has the smallest population of all the members of the Council of Europe. The country takes its name from a stonemason originating from the Roman colony on the island of Rab, in modern-day Croatia. In 257 CE Marinus participated in the reconstruction of Rimini's city walls by Liburnian pirates. The country is considered to have the earliest written governing documents still in effect. The country's economy mainly relies on finance, services and tourism. It is one of the wealthiest countries in the world in terms of GDP, with a figure comparable to the most developed European regions. San Marino is considered to have a highly stable economy, with one of the lowest unemployment rates in Europe, a budget surplus. It is the only country with more vehicles than people. Saint Marinus went to the city of Rimini as a stonemason. The official date of the founding of what is now known as the Republic is September 301. In 1631, its independence was recognized by the Papacy. The offer was declined by the Regents, fearing future retaliation from other states' revanchism. In recognition of this support, Giuseppe Garibaldi accepted the wish of San Marino not to be incorporated into the Italian state.
San Marino
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The San Marino constitution of 1600
San Marino
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Flag
San Marino
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The front passes Mount Titano in September 1944.
San Marino
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Mount Titano
198.
Sierra Leone
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Sierra Leone, officially the Republic of Sierra Leone, is a country in West Africa. It is bordered by Guinea on the north, Liberia in the south-east, the Atlantic Ocean in the south-west. Sierra Leone has a tropical climate, with a diverse environment ranging from savannah to rainforests. Sierra Leone has a total area of 71,740 km2 and a population of 7,075,641. Sierra Leone is divided into four geographical regions: the Western Area, which are subdivided into fourteen districts. Freetown is its political centre. Bo is the second largest city. The major cities are Koidu Town. This proxy war left more than 50,000 people much over million people displaced as refugees in neighbouring countries. More recently, the 2014 Ebola outbreak overburdened the weak healthcare infrastructure, leading to more deaths from medical neglect than Ebola itself. It created a humanitarian crisis situation and a negative spiral of weaker economic growth. The country has an extremely low life expectancy at 57.8 years. About ethnic groups inhabit customs. The two largest and most influential are the Temne and the Mende people. The Temne are predominantly found in the north of the country, while the Mende are predominant in the south-east.
Sierra Leone
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Fragments of prehistoric pottery from Kamabai Rock Shelter
Sierra Leone
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Flag
Sierra Leone
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An 1835 illustration of liberated Africans arriving in Sierra Leone.
Sierra Leone
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The colony of Freetown in 1856
199.
Suriname
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Suriname, officially known as the Republic of Suriname, is a sovereign state on the northeastern Atlantic coast of South America. It is bordered by Brazil to the south. At just under 165,000 km2, it is the smallest country in South America. Suriname has a population of approximately 566,000, most of whom live on Paramaribo. Long inhabited by numerous cultures of indigenous tribes, Suriname was contested before coming under Dutch rule in the late 17th century. In 1954, the country became one of the constituent countries of the Kingdom of the Netherlands. Its indigenous peoples have been increasingly active in working to preserve their traditional habitats. Suriname is considered to be a culturally Caribbean country, is a member of the Caribbean Community. While Dutch is the official language of government, education, Sranan, an English-based creole language, is a widely used lingua franca. Suriname is the only territory outside Europe where Dutch is spoken by a majority of the population. The people of Suriname are among the most diverse in the world, spanning a multitude of linguistic groups. The name Suriname may derive from a Taino indigenous people called Surinen, who inhabited the area at the time of European contact. British settlers, who founded the European colony along the Suriname River, spelled the name as "Surinam". When the territory was taken over by the Dutch, it became part of a group of colonies known as Dutch Guiana. The official spelling of the country's English name was changed from "Surinam" to "Suriname" in January 1978, but "Surinam" can still be found in English.
Suriname
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Maroon village, Suriname River, 1955
Suriname
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Flag
Suriname
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Presidential Palace of Suriname
Suriname
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Waterfront houses in Paramaribo, 1955
200.
Postage stamps
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Postage stamp may also refer to a formatting artifact in the display of film or video: Windowbox. A postage stamp is a small piece of paper, purchased and displayed on an item of mail as evidence of payment of postage. They are sometimes a source of net profit to the issuing agency, especially when sold to collectors who will not actually use them for postage. Stamps are usually rectangular, but triangles or other shapes are occasionally used. The stamp is affixed to an envelope or other postal cover the customer wishes to send. This procedure marks the stamp as used to prevent its reuse. In modern usage, postmarks generally indicate the date and point of origin of the mailing. The mailed item is then delivered to the address the customer has applied to the envelope or parcel. Postage stamps have facilitated the delivery of mail since the 1840s. Before then, ink and hand-stamps, usually made from wood or cork, were often used to frank the mail and confirm the payment of postage. The first adhesive postage stamp, commonly referred to as the Penny Black, was issued in the United Kingdom in 1840. There are varying accounts of the inventor or inventors of the stamp. The postage stamp resolved this issue in a simple and elegant manner, with the additional benefit of room for an element of beauty to be introduced. Concurrently with the first stamps, the UK offered wrappers for mail. With the conveniences stamps offered, their use resulted in greatly increased mailings during the 19th and 20th centuries.
Postage stamps
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The main components of a stamp: 1. Image 2. Perforations 3. Denomination 4. Country name
Postage stamps
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Rowland Hill
Postage stamps
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The Penny Black, the world’s first postage stamp.
Postage stamps
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Rows of perforations in a sheet of postage stamps.
201.
Neal Stephenson
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Neal Town Stephenson is an American writer and game designer known for his works of speculative fiction. His novels have been categorized as science fiction, historical fiction, cyberpunk, baroque. Stephenson's work explores subjects such as mathematics, cryptography, linguistics, philosophy, the history of science. He also writes non-fiction articles about technology in publications such as Wired. He has also written novels with George Jewsbury, under the collective pseudonym Stephen Bury. He is currently Magic Leap's Chief Futurist. Her father was a biochemistry professor. Stephenson's family moved to Ames, Iowa. He graduated from Ames High School in 1977. He graduated with a B.A. in geography and a minor in physics. Since 1984, Stephenson currently lives in Seattle with his family. Zodiac, was a thriller following the exploits of a radical environmentalist protagonist in his struggle against corporate polluters. Neither novel showcased concerns that Stephenson would further develop in his later work. Snow Crash was the first of Stephenson's epic fiction novels. Stephenson's next novel, published in 1995, was The Diamond Age: or A Young Lady's Illustrated Primer, which introduced many of today's real world technological discoveries.
Neal Stephenson
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Stephenson at Science Foo Camp 2008
Neal Stephenson
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Discussing Anathem at MIT in 2008
Neal Stephenson
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Stephenson at the Starship Century Symposium at UCSD in 2013
Neal Stephenson
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Stephenson at the National Book Festival in 2004
202.
Anathem
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Anathem is a speculative fiction novel by Neal Stephenson, published in 2008. Major themes include the philosophical debate between Platonic realism and nominalism. Anathem is set around the fictional planet Arbre. Thousands of years before the events in the planet's intellectuals entered concents to protect their activities from the collapse of society. Fraa Erasmas, is an avout at the Concent of Saunt Edhar. Orolo, discovers that an alien spacecraft is orbiting Arbre -- a fact that the Sæcular Power attempts to cover up. Orolo secretly observes the alien ship with technology, prohibited for the avout. Erasmas becomes aware of the content of Orolo's research after Orolo is banished within the concent. But the presence of the alien ship soon becomes an open secret among many of the avout at St. Edhar. The Sæcular Power calls forth many avout from Saunt Edhar, including Erasmas, along with one Millenarian – Fraa Jad. Several companions, including Fraa Jad, decide to seek out Orolo. After a dangerous journey over the planet's frozen pole, his comrades eventually arrive at a concent-like establishment called Orithena, reunite with Fraa Orolo. During the discussions between Orolo and Erasmas, a small spacecraft lands on Orithena. Dead of a recent gunshot wound. She brings with her four vials of blood – presumably that of the aliens – and much evidence about their technology.
Anathem
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Roger Penrose inspired the novel's "Teglon tiles", based on the aperiodic Penrose tiles, and the discussion of the brain as a quantum computer, based on Penrose's The Emperor's New Mind.
Anathem
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Cover of the hardcover first edition, featuring an analemma behind the author's name
203.
British flag theorem
British flag theorem
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According to the British flag theorem, the red squares have the same total area as the blue squares
204.
Pons asinorum
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This statement is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. Euclid's proof involves drawing auxiliary lines to these extensions. There has been much debate as to why, given that it makes the proof more complicated, Euclid added the second conclusion to the theorem. The proof relies heavily on what is today called side-angle-side, the previous proposition in the Elements. Proclus' variation of Euclid's proof proceeds as follows: Let ABC AC being the equal sides. Construct E on AC so that AD = AE. Draw BE, DC and DE. Therefore angle ABE = angle ACD, angle ADC = angle AEB, BE=CD. Since AD = AE, BD = CE by subtraction of equal parts. Therefore angle BDE = angle CED and angle BED = angle CDE. Since angle BDE = angle CED and angle CDE = angle BED, angle BDC = angle CEB by subtraction of equal parts. In particular, angle CBD = BCE, to be proved. Proclus gives a much shorter proof attributed to Pappus of Alexandria. It requires no additional construction at all.
Pons asinorum
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The pons asinorum in Byrne's edition of the Elements showing part of Euclid's proof.
205.
Fermat's Last Theorem
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The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity. The successful proof was released by Andrew Wiles, formally published by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved in mathematics for the following three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n = 4 proved, it suffices to prove the theorem for exponents n that are prime numbers. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. The solution came in a roundabout manner, from a completely different area of mathematics. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was widely considered completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat's Last Theorem. This potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermat's Last Theorem that depended on the modularity theorem.
Fermat's Last Theorem
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The 1670 edition of Diophantus ' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio Domini Petri de Fermat).
Fermat's Last Theorem
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Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the margin that was too small to contain Fermat's alleged proof of his "last theorem".
Fermat's Last Theorem
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British mathematician Andrew Wiles
206.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, initially listed as an advancement in geodesy. The study of algebra first emerged in the mid-1800s.
Linear algebra
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The three-dimensional Euclidean space R 3 is a vector space, and lines and planes passing through the origin are vector subspaces in R 3.
207.
Lp space
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In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, of topological vector spaces. Lebesgue spaces have applications in physics, statistics, other disciplines. In penalized regression, ` penalty' refer to penalizing either the L1 norm of a solution's vector of parameter values, or its L2 norm. Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero. Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Net regularization uses a term, a combination the L1 norm and the L2 norm of the parameter vector. The Fourier transform for the real line, maps Lp to Lq, where 1 ≤ p ≤ 2 and 1/p + 1/q = 1. This is a consequence of the Riesz–Thorin interpolation theorem, is made precise with the Hausdorff–Young inequality. By contrast, if p > 2, the Fourier transform does not map into Lq. Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ2, where E is a set with an appropriate cardinality. The Euclidean distance between two points x and y is the length ||x − y||2 of the straight line between the two points.
Lp space
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Unit circle (superellipse) in p = 3 / 2 norm
208.
Parallelogram law
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In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. Using the notation in the diagram on the right, the sides are. It can be seen from the diagram that, for a parallelogram, x = 0, the general formula simplifies to the parallelogram law. In an inner product space, the norm is determined using the inner product: ∥ x ∥ 2 = ⟨ x, x ⟩. Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms. Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p-norm if and only if p = 2, the so-called Euclidean norm or standard norm. For any norm satisfying the parallelogram law, the inner product generating the norm is unique as a consequence of the polarization identity. Commutative property Inner product space Normed vector space Polarization identity Weisstein, Eric W. "Parallelogram Law". MathWorld.
Parallelogram law
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A parallelogram. The sides are shown in blue and the diagonals in red.
209.
Ptolemy's theorem
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In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after mathematician Ptolemy. Ptolemy used the theorem to creating his table of chords, a trigonometric table that he applied to astronomy. In the context of geometry, the above equality is often simply written as AC·BD=AB·CD+BC·AD. Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. Proof: Follows immediately from Ptolemy's theorem: q s = p s + r s ⇒ q = p + r. Any square can be inscribed in a circle whose center is the center of the square. More generally, if the quadrilateral is a rectangle with sides a and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d2, the right side of Ptolemy's relation is the sum a2 + b2. A more interesting example is the relation between the length a of the length b of the 5 chords in a regular pentagon. In this case the relation reads b2 = + ab which yields the golden ratio φ = b a = 1 + 5 2. ⇒ c = 2 φ. Whence the side of the inscribed decagon is obtained in terms of the diameter.
Ptolemy's theorem
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Corollary 1: Pythagoras' Theorem
Ptolemy's theorem
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Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral.
210.
Pythagorean tiling
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Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. This tiling has rotational symmetry around each of its squares. Generalizations of this tiling to three dimensions have also been studied. The Pythagorean tiling is the unique tiling by squares of two different sizes, both equitransitive. Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by regular octagons. However, the two tilings have different sets of symmetries, because the square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isn't. It is a chiral pattern, meaning that it is impossible to superpose it on top of its image using only translations and rotations. If this requirement is relaxed then there are eight additional uniform tilings. Three are formed from equilateral triangles and regular hexagons. The remaining one is the Pythagorean tiling. Similarly, may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares. Although the Pythagorean tiling is itself periodic its cross sections can be used to generate aperiodic sequences. In this sequence, the relative proportion of 1s will be in the ratio x:1. This proportion can not be achieved by a periodic sequence of 1s, because it is irrational, so the sequence is aperiodic.
Pythagorean tiling
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Street Musicians at the Doorway of a House, Jacob Ochtervelt, 1665. As observed by Nelsen the floor tiles in this painting are set in the Pythagorean tiling
Pythagorean tiling
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A Pythagorean tiling
Pythagorean tiling
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Periodic
Pythagorean tiling
211.
Rational trigonometry
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His ideas are set out in Divine Proportions: Rational Trigonometry to Universal Geometry. Rational trigonometry avoids direct use of transcendental functions like cosine by substituting their squared equivalents. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature. Rational trigonometry follows an approach built to the topics of elementary geometry. ` angle' is replaced with the squared value of the usual sine ratio associated to either angle between two lines.. . Following this, it is claimed, makes many classical results of Euclidean geometry applicable in rational form over any field not of characteristic two. The book Divine Proportions shows the application of calculus using trigonometric functions, including three-dimensional volume calculations. It also deals to situations involving irrationals, such as the proof that Platonic Solids all have rational ` spreads' between their faces. Distance both measure separation of points in Euclidean space. Spread gives measure to the separation of two lines as a single dimensionless number in the range for Euclidean geometry. It has several differences from angle, discussed in the section below. Spread can have several interpretations. Trigonometric: it is the sine-ratio for the quadrances in a right triangle and therefore equivalent to the square of the sine of the angle. Vector: as a rational function of the relative directions of a pair of lines where they meet.
Rational trigonometry
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Suppose ℓ 1 and ℓ 2 intersect at the point A. Let C be the foot of the perpendicular from B to ℓ 2. Then the spread is s = Q / R.
212.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
213.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing back issues of academic journals, it now also includes books and primary sources, current issues of journals. It provides full-text searches of almost 2,000 journals. President of Princeton University from 1972 to 1988, founded JSTOR. Most libraries found it prohibitively expensive in terms of space to maintain a comprehensive collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term. Full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution. JSTOR originally encompassed ten economics and history journals. It became a fully searchable index accessible from any ordinary web browser. Special software was put in place to make graphs clear and readable. With the success of this limited project, then-president of JSTOR, wanted to expand the number of participating journals. The work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially. Until January 2009 JSTOR operated in New York City and in Ann Arbor, Michigan.
JSTOR
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The JSTOR front page
214.
ArXiv
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In many fields of physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 1991, arXiv.org passed the half-million article milestone on October 3, 2008, hit a million by the end of 2014. By 2014 the rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX format, which allowed scientific papers to be easily transmitted over the Internet and rendered client-side. The number of papers being sent soon filled mailboxes to capacity. Additional modes of access were soon added the World Wide Web in 1993. The e-print was quickly adopted to describe the articles. Its original name was xxx.lanl.gov. It is now hosted principally with 8 mirrors around the world. Its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Scientists regularly upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. Annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. In September 2011, Cornell University Library took overall financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a sentence".
ArXiv
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arXiv
ArXiv
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A screenshot of the arXiv taken in 1994, using the browser NCSA Mosaic. At the time, HTML forms were a new technology.
215.
American Mathematical Society
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The society is one of the four parts of a member of the Conference Board of the Mathematical Sciences. Fiske became secretary. The society soon ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The facto journal, as intended, was influential in increasing membership. In 1891 Charlotte Scott became the first woman to join the society. In 1951, the society's headquarters moved to Providence, Rhode Island. The society later added an office in 1992. Mary W. Gray challenged that situation by "sitting in on the Council meeting in Atlantic City. When she was told she had to leave, she refused saying she would wait until the police came. The process of democratization of the Society had begun." Julia Robinson was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, with the intent of being the flagship journal of the AMS. The 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four regional sections of the AMS hold meetings in the fall of each year.
American Mathematical Society
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American Mathematical Society
216.
Google Books
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Books are provided either by publishers and authors, through the Google Books Partner Program, or by Google's library partners, through the Library Project. Additionally, Google has partnered with a number of magazine publishers to digitize their archives. The Publisher Program was first known as'Google Print' when it was introduced at the Frankfurt Book Fair in October 2004. The Google Books Library Project, which scans works in the collections of library partners and adds them to the digital inventory, was announced in December 2004. The scanning process has slowed down in academic libraries. Google estimated in 2010 that there were about 130 million distinct titles in the world, stated that it intended to scan all of them. Results from Google Books show up in both the universal Google Search well as in the dedicated Google Books website. If Google believes the book is still under copyright, a user sees "snippets" of text around the queried search terms. All instances of the search terms in the book text appear with a yellow highlight. In-print books acquired through the Partner Program are also available for full view if the publisher has given permission, although this is rare. Usually, the publisher can set the percentage of the book available for preview. Users are restricted from printing book previews. A watermark reading "Copyrighted material" appears at the bottom of pages. All books acquired through the Partner Program are available for preview. This could be because Google cannot identify the owner or the owner declined permission.
Google Books
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Formats
Google Books
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Google Books screenshot
217.
Education Resources Information Center
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The Education Resources Information Center is an online digital library of education research and information. ERIC is sponsored by the Institute of Education Sciences of the United States Department of Education. Education information are essential to improving teaching, learning, educational decision-making. ERIC provides access with hundreds of new records added every week. A key component of ERIC is its collection of grey literature in education, largely available in full text in Adobe PDF format. Approximately one quarter of the complete ERIC Collection is available in full text. Materials with no full text available can often be accessed library holdings. Users can also access the collection through commercial database vendors, Internet search engines. To help users find the information they are seeking, ERIC produces the Thesaurus of ERIC Descriptors. This is a carefully selected list of education-related phrases used to tag materials by subject and make them easier to retrieve through a search. Prior to January 2004, the ERIC network consisted of sixteen subject-specific clearinghouses, three support components. Paper-based processes converted to electronic, thus streamlining operations and speeding delivery of content. ERIC website ERIC Digests, a repository for materials produced by the former ERIC Clearinghouse system up to 2003
Education Resources Information Center
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Logo of ERIC
218.
Institute of Education Sciences
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The Institute of Education Sciences is the independent, non-partisan statistics, research, evaluation arm of the U.S. Department of Education. It was created as part of 2002. The first director of IES was Dr. Grover J. "Russ" Whitehurst, served for six years. Dr. Ruth Neild is currently delegated the duties of the director. Dr. Joy Lesnick is the acting commissioner of NCEE. NCER also supports training programs to prepare researchers to conduct scientific education research. Dr. Thomas W. Brock is the commissioner of NCER. National Center for Education Statistics —NCES is the primary federal entity that collects and analyzes data related to education in the United States and other nations. Among the initiatives that NCES oversees is the National Assessment of Educational Progress. Dr. Peggy G. Carr is the acting commissioner of NCES. NCSER also supports training programs to prepare researchers to conduct scientific special education research. Dr. Joan E. McLaughlin is the commissioner of NCSER. The Board consults with the director and the commissioners to identify research and organizational priorities for IES. Dean of the Annette Caldwell Simmons School of Education and Human Development at Southern Methodist University, is the chair of the Board.
Institute of Education Sciences
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Secretary of Education
219.
U.S. Department of Education
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The United States Department of Education, also referred to as the ED for Education Department, is a Cabinet-level department of the United States government. The Department of Education is administered by the United States Secretary of Education. It is by far the smallest Cabinet-level department, with about 5,000 employees. It has an annual budget of US$ Billion. The agency's official abbreviation is "ED", not "DOE", which refers to the United States Department of Energy. It is also often abbreviated informally as "DoED". A previous Department of Education was soon demoted to an Office in 1868. As an agency not represented in the president's cabinet, it quickly became a relatively minor bureau in the Department of the Interior. In 1939, the bureau was transferred to the Federal Security Agency, where it was renamed the Office of Education. In 1953, the Federal Security Agency was upgraded as the Department of Health, Education, Welfare. In 1979, President Carter advocated for creating a cabinet-level Department of Education. Carter's plan was to transfer most of Welfare's education-related functions to the Department of Education. The National Education Association supported the bill, while the American Federation of Teachers opposed it. As of 1979, the Office of Education had an annual budget of $12 billion. Congress appropriated to the Department of Education 17,000 employees when establishing the Department of Education.
U.S. Department of Education
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The Lyndon Baines Johnson Department of Education building in 2006
U.S. Department of Education
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Seal of the U.S. Department of Education
U.S. Department of Education
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A construction project to repair and update the building façade at the Department of Education headquarters in 2002 resulted in the installation of structures at all of the entrances to protect employees and visitors from falling debris. ED redesigned these protective structures to promote the No Child Left Behind Act. The structures were temporary and were removed in 2008. Source: U.S. Department of Education,
220.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available on CD-ROM. The presentation is technical in nature. The encyclopedia was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains three-dimensional objects. A dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between the European Mathematical Society. All entries will be monitored by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989.
Encyclopedia of Mathematics
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Encyclopedia of Mathematics snap shot
Encyclopedia of Mathematics
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A complete set of Encyclopedia of Mathematics at a university library.
221.
Java (programming language)
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Java is a general-purpose computer programming language, concurrent, class-based, object-oriented, specifically designed to have as few implementation dependencies as possible. Java applications are typically compiled to bytecode that can run on any Java virtual machine regardless of computer architecture. As of 2016, Java is one of the most popular programming languages in use, particularly for client-server web applications, with a reported 9 million developers. Java was originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++, but it has fewer low-level facilities than either of them. The original and reference implementation Java compilers, virtual machines, class libraries were originally released by Sun under proprietary licences. Others have also developed alternative implementations of these Sun technologies, such as the GNU Compiler for Java, GNU Classpath, IcedTea-Web. Patrick Naughton initiated the Java project in June 1991. It was too advanced for the digital industry at the time. The language was initially called Oak after an oak tree that stood outside Gosling's office. Later the project went by the name Green and was finally renamed Java, from Java coffee. Gosling designed Java with a C/C++-style syntax that system and application programmers would find familiar. Sun Microsystems released the first public implementation as Java 1.0 in 1995. It promised "Write Once, Run Anywhere", providing no-cost run-times on popular platforms. Fairly secure and featuring configurable security, it allowed network- and file-access restrictions.
Java (programming language)
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James Gosling, the creator of Java (2008)
Java (programming language)
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Java
Java (programming language)
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Java Control Panel, version 7
222.
Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly increasingly also by archives and museums. The GND is managed with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license. The GND specification provides a hierarchy of high-level sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format.
Integrated Authority File
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GND screenshot
223.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established for the purpose of assisting members of the National Diet of Japan in researching matters of public policy. The library is similar in scope to the United States Library of Congress. The National Diet Library consists of several other branch libraries throughout Japan. Its need for information was "correspondingly small." The original Diet libraries "never developed either the services which might have made them vital adjuncts of genuinely responsible legislative activity." Until Japan's defeat, moreover, the executive had controlled all political documents, depriving the Diet of access to vital information. In 1946, each house of the Diet formed its own National Diet Library Standing Committee. Hani envisioned the new body as "both a ` citadel of popular sovereignty," and the means of realizing a "peaceful revolution." The National Diet Library opened with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori. The philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL became the only national library in Japan. At this time the collection gained an additional million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, adjacent to the National Diet.
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
224.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including algebraic proofs, with some dating back thousands of years. He may well have been the first to prove it. In any event, the proof is called a proof by rearrangement. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D. That Pythagoras originated this very simple proof is sometimes inferred from the writings of mathematician Proclus. This is known as the Pythagorean one. If the length of b are known, then c can be calculated as c = a 2 + b 2. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; The Pythagorean Proposition contains 370 proofs.
Pythagorean theorem
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The Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
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Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagorean theorem
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Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
Pythagorean theorem
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Exhibit on the Pythagorean theorem at the Universum museum in Mexico City