1.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of 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 area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all 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. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
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Circumscribed cylinder to a sphere
Sphere
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A two-dimensional perspective projection of a sphere
Sphere
Sphere
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Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
2.
History of geometry
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Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of mathematics, the other being the study of numbers. Classic geometry was focused in compass and straightedge constructions, geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Among these were some surprisingly sophisticated principles, and a mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B. C. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle and this assumes that π is 4×², with an error of slightly over 0.63 percent. Problem 48 involved using a square with side 9 units and this square was cut into a 3x3 grid. The diagonal of the squares were used to make an irregular octagon with an area of 63 units. This gave a value for π of 3.111. The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula, the Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3, the Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3, the Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a used for measuring the travel of the Sun, therefore. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did, the Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts on this include the Satapatha Brahmana and the Śulba Sūtras. According to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, the diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately
History of geometry
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Part of the " Tab.Geometry. " (Table of Geometry) from the 1728 Cyclopaedia.
History of geometry
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Rigveda manuscript in Devanagari.
History of geometry
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Statue of Euclid in the Oxford University Museum of Natural History.
History of geometry
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Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310)
3.
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. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, 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. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, 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 numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
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.
4.
Synthetic geometry
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Synthetic geometry is the study of geometry without the use of coordinates or formulas. It relies on the method and the tools directly related to them. Only after the introduction of methods was there a reason to introduce the term synthetic geometry to distinguish this approach to geometry from other approaches. Other approaches to geometry are embodied in analytic and algebraic geometries, geometry, as presented by Euclid in the elements, is the quintessential example of the use of the synthetic method. It was the method of Isaac Newton for the solution of geometric problems. Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry, for example the geometer Jakob Steiner hated analytic geometry, and always gave preference to synthetic methods. The process of logical synthesis begins with some arbitrary but definite starting point and this starting point is the introduction of primitive notions or primitives and axioms about these primitives, Primitives are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are such as points, lines and planes. Axioms are statements about these primitives, for example, any two points are incident with just one line. Axioms are assumed true, and not proven and they are the building blocks of geometric concepts, since they specify the properties that the primitives have. From a given set of axioms, synthesis proceeds as a carefully constructed logical argument, when a significant result is proved rigorously, it becomes a theorem. There is no fixed set for geometry, as more than one consistent set can be chosen. Each such set may lead to a different geometry, while there are examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of geometry in the singular, historically, Euclids parallel postulate has turned out to be independent of the other axioms. Simply discarding it gives absolute geometry, while negating it yields hyperbolic geometry, other consistent axiom sets can yield other geometries, such as projective, elliptic, spherical or affine geometry. Axioms of continuity and betweeness are also optional, for example, following the Erlangen program of Klein, the nature of any given geometry can be seen as the connection between symmetry and the content of the propositions, rather than the style of development. One of the early French analysts summarized synthetic geometry this way, for example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory than is found by starting with a vector space of dimension three
Synthetic geometry
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Projecting a sphere to a plane.
5.
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. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation, analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
Analytic geometry
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Cartesian coordinates
6.
Riemannian geometry
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This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other 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 and it is a very broad and abstract generalization of the differential geometry of surfaces in R3. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century and it deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of Non-Euclidean geometry. Any smooth manifold admits a Riemannian metric, which helps to solve problems of differential topology. It also serves as a level for the more complicated structure of pseudo-Riemannian manifolds. 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. Dislocations and Disclinations produce torsions and curvature, the choice is made depending on its importance and elegance of formulation. Most of the results can be found in the monograph by Jeff Cheeger. The formulations given are far from being very exact or the most general and this list is oriented to those who already know the basic definitions and want to know what these definitions are about. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ where χ denotes the Euler characteristic of M and this theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. Nash embedding theorems also called fundamental theorems of Riemannian geometry and they state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. If M is a connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere. Given constants C, D and V, there are finitely many compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D. There is an εn >0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K| ≤ εn, G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture, M is diffeomorphic to Rn if it has positive curvature at only one point. There is a constant C = C such that if M is a compact connected n-dimensional Riemannian manifold with sectional curvature then the sum of its Betti numbers is at most C. Given constants C, D and V, there are finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D. It implies that any two points of a connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic
Riemannian geometry
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Bernhard Riemann
Riemannian geometry
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Projecting a sphere to a plane.
7.
Symplectic geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds, that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the space of certain classical systems takes on the structure of a symplectic manifold. A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold, on this space is defined a geometric object, the symplectic form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays an analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures areas, Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the position q and the p, which form a point in the Euclidean plane ℝ2. In this case, the form is ω = d p ∧ d q and is an area form that measures the area A of a region S in the plane through integration. The area is important because as conservative dynamical systems evolve in time, higher dimensional symplectic geometries are defined analogously. Symplectic geometry has a number of similarities with and differences from Riemannian geometry, unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. Another difference with Riemannian geometry is not every differentiable manifold need admit a symplectic form. For example, every symplectic manifold is even-dimensional and orientable, both concepts play a fundamental role in their respective disciplines. Every Kähler manifold is also a symplectic manifold, most symplectic manifolds, one can say, are not Kähler, and so do not have an integrable complex structure compatible with the symplectic form. These invariants also play a key role in string theory, Symplectic geometry is also called symplectic topology although the latter is really a subfield concerned with important global questions in symplectic geometry. The term symplectic is a calque of complex, introduced by Weyl, previously and this naming reflects the deep connections between complex and symplectic structures. ISBN 2-88124-901-9. de Gosson, Maurice A. Symplectic Geometry, bulletin of the American Mathematical Society. Hazewinkel, Michiel, ed. Symplectic structure, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Symplectic geometry
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Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.
8.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes, similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes, There are two main kinds of finite plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry, the simplest affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
9.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
Projective geometry
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Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
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Projecting a sphere to a plane.
Projective geometry
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Forms
10.
Compass-and-straightedge construction
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in 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, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its 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 segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
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
11.
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 the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
Curve
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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
12.
Orthogonal
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, and engineering. The word comes from the Greek ὀρθός, meaning upright, and γωνία, the ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle, in the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i. e. they form a right angle, two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y, two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a subspace is its orthogonal complement. Given a module M and its dual M∗, an element m′ of M∗, two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent, a set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set, nonzero pairwise orthogonal vectors are always linearly independent. In certain cases, the normal is used to mean orthogonal. For example, the y-axis is normal to the curve y = x2 at the origin, however, normal may also refer to the magnitude of a vector. In particular, a set is called if it is an orthogonal set of unit vectors. As a result, use of the normal to mean orthogonal is often avoided. The word normal also has a different meaning in probability and statistics, a vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality, in the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ. In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i. e. they make an angle of 90°, hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces
Orthogonal
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The line segments AB and CD are orthogonal to each other.
13.
Perpendicular
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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. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, for this reason, we may speak of two lines as being perpendicular without specifying an order. Perpendicularity easily extends to segments and rays, in symbols, A B ¯ ⊥ C D ¯ means line segment AB is perpendicular to line segment CD. A line is said to be perpendicular to an 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 instance of the more general mathematical concept of orthogonality, perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the perpendicular is sometimes used to describe much more complicated geometric orthogonality conditions. The word foot is used in connection with perpendiculars. This usage is exemplified in the top diagram, above, 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. Let Q and R be the points of intersection of two circles. Step 3, connect Q and R to construct the desired perpendicular PQ, to prove that the PQ is perpendicular to AB, use the SSS congruence theorem for and QPB to conclude that angles OPA and OPB are equal. Then use the SAS congruence theorem for triangles OPA and OPB to conclude that angles POA, to make the perpendicular to the line g at or through the point P using Thales theorem, see the animation at right. The Pythagorean Theorem can be used as the basis of methods of constructing right angles, for example, by counting links, three pieces of chain can be made with lengths in the ratio 3,4,5. These can be out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out gardens and fields, where the dimensions are large, the chains can be used repeatedly whenever required. If two lines are perpendicular to a third line, all of the angles formed along the third line are right angles
Perpendicular
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The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees.
14.
Parallel (geometry)
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
Parallel (geometry)
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As shown by the tick marks, lines a and b are parallel. This can be proved because the transversal t produces congruent corresponding angles, shown here both to the right of the transversal, one above and adjacent to line a and the other above and adjacent to line b.
15.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
Vertex (geometry)
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A vertex of an angle is the endpoint where two line segments or rays come together.
16.
Congruence (geometry)
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In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
Congruence (geometry)
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The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
17.
Similarity (geometry)
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Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling and this means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other zoomed in or out at some level. For example, all circles are similar to other, all squares are similar to each other. On the other hand, ellipses are not all similar to other, rectangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure and it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem, due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of which is necessary and sufficient for two triangles to be similar,1, the triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is, If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′, all the corresponding sides have lengths in the same ratio, AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle is an enlargement of the other, two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance, AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′ and this is known as the SAS Similarity Criterion. When two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′, there are several elementary results concerning similar triangles in Euclidean geometry, Any two equilateral triangles are similar. Two triangles, both similar to a triangle, are similar to each other. Corresponding altitudes of similar triangles have the ratio as the corresponding sides. Two right triangles are similar if the hypotenuse and one side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, the statement that the point F satisfying this condition exists is Walliss Postulate and is logically equivalent to Euclids Parallel Postulate
Similarity (geometry)
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Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log 2 3, which is approximately 1.58. (from Hausdorff dimension.)
Similarity (geometry)
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Figures shown in the same color are similar
18.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
Symmetry
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Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
Symmetry
Symmetry
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Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Symmetry
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The ceiling of Lotfollah mosque, Isfahan, Iran has 8-fold symmetries.
19.
Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. 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 sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
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historical image – create a line segment (1699)
20.
Ray (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
Ray (geometry)
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The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place).
21.
Two-dimensional space
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In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons
Two-dimensional space
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Bi-dimensional Cartesian coordinate system
22.
Altitude (triangle)
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In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the side is called the extended base of the altitude. The intersection between the base and the altitude is called the foot of the altitude. The length of the altitude, often called the altitude, is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude of that vertex and it is a special case of orthogonal projection. Altitudes can be used to compute the area of a triangle, one half of the product of an altitudes length, 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 and 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. If we denote the length of the altitude by hc, we then have the relation h c = p q For acute, 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. The product of the distances from the orthocenter to a vertex and this product is the squared radius of the triangles polar circle. The orthocenter H, the centroid G, the circumcenter O, and the center N of the nine-point circle all lie on a single line, known as the Euler line. The orthocenter is closer to the incenter I than it is to the centroid, the isogonal conjugate and also the complement of the orthocenter is the circumcenter. Four points in the plane such that one of them is the orthocenter of the triangle formed by the three are called an orthocentric system or orthocentric quadrangle. Let A, B, C denote the angles of the reference triangle, and let a = |BC|, b = |CA|, c = |AB| be the sidelengths. In the complex plane, let the points A, B and C represent the numbers z A, z B and respectively z C and assume that the circumcenter of triangle A B C is located at the origin of the plane. Then, the number z H = z A + z B + z C is represented by the point H
Altitude (triangle)
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Three altitudes intersecting at the orthocenter
23.
Parallelogram
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In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length, 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, rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles Rectangle – A parallelogram with four angles of equal size. Rhombus – A parallelogram with four sides of equal length, square – A parallelogram with four sides of equal length and angles of equal size. A simple quadrilateral is a if and only if any one of the following statements is true. Two pairs of opposite angles are equal in measure, one pair of opposite sides are parallel and equal in length. 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. Thus all parallelograms have all the properties listed above, and conversely, if just one of statements is true in a simple quadrilateral. Opposite sides of a parallelogram are parallel and so will never intersect, the area of a parallelogram is twice the area of a triangle created by one of its diagonals. The area of a parallelogram is also equal to the magnitude of the cross product of two adjacent sides. Any line through the midpoint of a parallelogram bisects the area, any non-degenerate affine transformation takes a parallelogram to another parallelogram. A parallelogram has rotational symmetry of order 2, if it also has exactly two lines of reflectional symmetry then it must be a rhombus or an oblong. If it has four lines of symmetry, it is a square. The perimeter of a parallelogram is 2 where a and b are the lengths of adjacent sides, unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area, the diagonals of a parallelogram divide it into four triangles of equal area. All of the formulas for general convex quadrilaterals apply to parallelograms
Parallelogram
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This parallelogram is a rhomboid as it has no right angles and unequal sides.
24.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
Square
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A regular quadrilateral (tetragon)
25.
Rhomboid
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Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of length is a rhombus but not a rhomboid. A parallelogram with right angled corners is a rectangle but not a rhomboid, the term rhomboid is now more often used for a parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids and this solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning, and let quadrilaterals other than these be called trapezia. Heath suggests that rhomboid was a term already in use. The rhomboid has no line of symmetry, but it has symmetry of order 2. In biology, rhomboid may describe a geometric rhomboid or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, in a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope
Rhomboid
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These shapes are rhomboids
26.
Trapezoid
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The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, the first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclids Elements. This article uses the term trapezoid in the sense that is current in the United States, in many 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 in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute, an acute trapezoid is also an isosceles trapezoid, if its sides have the same length, and the base angles have the same measure. An obtuse trapezoid with two pairs of sides is a parallelogram. A parallelogram has central 2-fold 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 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. Some define a trapezoid as a quadrilateral having one pair of parallel sides. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, the latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined and this article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals, under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges, rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Four lengths a, c, b, d can constitute the sides of a non-parallelogram trapezoid with a and b parallel only when | d − c | < | b − a | < d + c. The quadrilateral is a parallelogram when d − c = b − a =0, the angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio, the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar. The diagonals cut the quadrilateral into four triangles of which one pair have equal areas
Trapezoid
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The Temple of Dendur in the Metropolitan Museum of Art in New York City
Trapezoid
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Trapezoid
Trapezoid
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Example of a trapeziform pronotum outlined on a spurge bug
27.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the 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 segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, 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, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
Circle
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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
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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.
28.
Diameter
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In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
Diameter
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Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
29.
Cube
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Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the lungs and the bloodstream, the International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen, beryllium copper is a non-ferrous alloy used in springs, spring wire, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
Cube
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Example of a non-sparking tool made of beryllium copper
Cube
30.
Cylinder (geometry)
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Cylinder (geometry)
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Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
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A right circular cylinder with radius r and height h.
Cylinder (geometry)
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In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
31.
Alhazen
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Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, also known by the Latinization Alhazen or Alhacen, was an Arab Muslim scientist, mathematician, astronomer, and philosopher. Ibn al-Haytham made significant contributions to the principles of optics, astronomy, mathematics and he was the first to explain that vision occurs when light bounces on an object and then is directed to ones eyes. He spent most of his close to the court of the Fatimid Caliphate in Cairo and earned his living authoring various treatises. In medieval Europe, Ibn al-Haytham was honored as Ptolemaeus Secundus or simply called The Physicist and he is also sometimes called al-Baṣrī after his birthplace Basra in Iraq, or al-Miṣrī. Ibn al-Haytham was born c.965 in Basra, which was part of the Buyid emirate. Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040. Legend has it that after deciding the scheme was impractical and fearing the caliphs anger, during this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy. Among his students were Sorkhab, a Persian from Semnan who was his student for three years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen. Alhazen made significant contributions to optics, number theory, geometry, astronomy, Alhazens work on optics is credited with contributing a new emphasis on experiment. In al-Andalus, it was used by the prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text. A Latin translation of the Kitab al-Manazir was made probably in the twelfth or early thirteenth century. His research in catoptrics centred on spherical and parabolic mirrors and spherical aberration and he made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the known as Alhazens problem. Alhazen wrote as many as 200 books, although only 55 have survived, some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew, the crater Alhazen on the Moon is named in his honour, as was the asteroid 59239 Alhazen. In honour of Alhazen, the Aga Khan University named its Ophthalmology endowed chair as The Ibn-e-Haitham Associate Professor, Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10, 000-dinar banknote issued in 2003, and on 10-dinar notes from 1982. The 2015 International Year of Light celebrated the 1000th anniversary of the works on optics by Ibn Al-Haytham, Alhazens most famous work is his seven-volume treatise on optics Kitab al-Manazir, written from 1011 to 1021. Optics was translated into Latin by a scholar at the end of the 12th century or the beginning of the 13th century
Alhazen
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Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen's Book of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Alhazen
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Alhazen (Ibn al-Haytham)
Alhazen
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The theorem of Ibn Haytham
Alhazen
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Alhazen on Iraqi 10 dinars
32.
Apollonius of Perga
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Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic and his definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. Apollonius worked on other topics, including astronomy. Most of the work has not survived except in references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, for such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states, “Apollonius, the geometrician. Came from Perga in Pamphylia in the times of Ptolemy Euergetes, the ruins of the city yet stand. It was a center of Hellenistic culture, Euergetes, “benefactor, ” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC, times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain, the approximate times of Apollonius are thus certain, but no exact dates can be given. The figure Specific birth and death years stated by the scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt, never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. Someone designated “of Perga” might well be expected to have lived and worked there, to the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria, philip was assassinated in 336 BC. Alexander went on to fulfill his plan by conquering the vast Iranian empire, the material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” and he intended to verify and emend the books, releasing each one as it was completed
Apollonius of Perga
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Pages from the 9th century Arabic translation of the Conics
Apollonius of Perga
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Parabola connection with areas of a square and a rectangle, that inspired Apollonius of Perga to give the parabola its current name.
33.
Michael Atiyah
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Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng is an English mathematician specialising in geometry. Atiyah grew up in Sudan and Egypt and spent most of his life in the United Kingdom at Oxford and Cambridge. He has been president of the Royal Society, master of Trinity College, Cambridge, chancellor of the University of Leicester, since 1997, he has been an honorary professor at the University of Edinburgh. Atiyahs mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations. Some of his recent work was inspired by theoretical physics, in particular instantons and monopoles. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, Atiyah was born in Hampstead, London, to a Lebanese father, the academic, Eastern Orthodox, Edward Atiyah and Scot Jean Atiyah. Patrick Atiyah is his brother, he has one brother, Joe. He returned to England and Manchester Grammar School for his HSC studies and his undergraduate and postgraduate studies took place at Trinity College, Cambridge. He was a student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. Atiyah married Lily Brown on 30 July 1955, with whom he has three sons, in 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherines College. He became Savilian Professor of Geometry and a fellow of New College, Oxford. He was president of the London Mathematical Society from 1974 to 1976, Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002. He also contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies, within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director. He was President of the Royal Society, Master of Trinity College, Cambridge, Chancellor of the University of Leicester, since 1997, he has been an honorary professor in the University of Edinburgh. Atiyah has collaborated with other mathematicians. His later research on gauge field theories, particularly Yang–Mills theory, other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut. Atiyah said that the mathematician he most admired was Hermann Weyl, the six volumes of Atiyahs collected papers include most of his work, except for his commutative algebra textbook and a few works written since 2004
Michael Atiyah
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Michael Atiyah in 2007.
Michael Atiyah
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Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master
Michael Atiyah
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The Institute for Advanced Study in Princeton, where Atiyah was professor from 1969 to 1972
Michael Atiyah
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The Mathematical Institute in Oxford, where Atiyah supervised many of his students
34.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Coxeter
35.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several 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 year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 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, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
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
36.
Mikhail Leonidovich Gromov
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Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University, Gromov has won several prizes, including the Abel Prize in 2009 for his revolutionary contributions to geometry. Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings, disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel and he changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math
Mikhail Leonidovich Gromov
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Mikhail Gromov
37.
Nikolai Lobachevsky
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Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry. William Kingdon Clifford called Lobachevsky the Copernicus of Geometry due to the character of his work. He was one of three children and his father, a clerk in a land surveying office, died when he was seven, and his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University, at Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of German mathematician Carl Friedrich Gauss. Lobachevsky received a degree in physics and mathematics in 1811. He served in administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva and they had a large number of children. He was dismissed from the university in 1846, ostensibly due to his health, by the early 1850s, he was nearly blind. He died in poverty in 1856, Lobachevskys main achievement is the development of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclids fifth postulate from other axioms, Euclids fifth is a rule in Euclidean geometry which states that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true and this idea was first reported on February 23,1826 to the session of the department of physics and mathematics, and this research was printed in the UMA in 1829–1830. The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry and he developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of geometry which has many applications. Hyperbolic geometry is referred to as Lobachevskian geometry or Bolyai–Lobachevskian geometry. Some mathematicians and historians have claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss. Gauss himself appreciated Lobachevskys published works very highly, but they never had personal correspondence between them prior to the publication, Lobachevskys magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry and he also wrote Geometrical Investigations on the Theory of Parallels and Pangeometry
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
38.
Minggatu
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Minggatu, full name Sharabiin Myangat was a Mongolian astronomer, mathematician, and 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 and his name first appeared in official Chinese records in 1713, among the Kangxi Emperors retinue, as a shengyuan of the Imperial Astronomical Bureau. He worked there at a time when Jesuit missionaries were in charge of calendar reforms and he also participated in the work of compiling and editing three very important books in astronomy and joined the team of Chinas area measurement. From 1724 up to 1759, he worked at the Imperial Observatory and he participated in drafting and editing the calendar and the study of the armillary sphere. He was the first person in China who calculated infinite series, in the 1730s, he first established and used what was later to be known as Catalan numbers. Minggatus work is remarkable in that expansions in series, trigonometric and 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 and it was due to his geographical surveys in Xinjiang that the Complete Atlas of the Empire was finished. From 1760-1763, shortly before his death, he was administrator of the Imperial Astronomical Bureau, in 1910, Japanese mathematician Yoshio Mikami mentioned that Minggatu was the first Chinese who had ever entered into the field of analytical research methods. On May 26,2002, the minor planet 28242 was named after Minggatu as 28242 Mingantu, the nomination ceremony and traditional meeting were held in Minggatus hometown in August 2002. More than 500 delegates and 20,000 local residents gathered together to celebrate, the Chinese government named Ming’s hometown as Ming Antu Town. Ming Antus infinite series expansion of trigonometric functions
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
39.
Pythagoras
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Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
Pythagoras
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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.
40.
Nasir al-Din al-Tusi
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Khawaja Muhammad ibn Muhammad ibn al-Hasan al-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, in Hamadan and Tus he studied the Quran, Hadith, Shia jurisprudence, logic, philosophy, mathematics, medicine and 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 who was killed by Mongol invaders. In Mosul he studied mathematics and astronomy with Kamal al-Din Yunus and he was captured after the invasion of the Alamut castle by the Mongol forces. Tusi has about 150 works, of which 25 are in Persian and the remaining are in Arabic, 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 fiilm al-hayah – 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, akhlaq-i Nasiri – A work on ethics. Al-Risalah al-Asturlabiyah – A Treatise on astrolabe, Zij-i ilkhani – A major astronomical treatise, completed in 1272. Sharh al-isharat Awsaf al-Ashraf a short work in Persian Tajrīd al-iʿtiqād – A commentary on Shia doctrines. During his stay in Nishapur, Tusi established a reputation as an exceptional scholar, tusi’s prose writing, which number over 150 works, represent one of the largest collections by a single Islamic author. Writing in both Arabic and Persian, Nasir al-Din Tusi dealt with religious topics and non-religious or secular subjects. His works include the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy, Autolycus, Tusi convinced Hulegu Khan to construct an observatory for establishing accurate astronomical tables for better astrological predictions. Beginning in 1259, the Rasad Khaneh observatory was constructed in Azarbaijan, south of the river Aras, and to the west of Maragheh, the capital of the Ilkhanate Empire. Based on the observations in this for the time being most advanced observatory and this book contains astronomical tables for calculating the positions of the planets and the names of the stars. His model for the system is believed to be the most advanced of his time. Between Ptolemy and Copernicus, he is considered by many to be one of the most eminent astronomers of his time, for his planetary models, he invented a geometrical technique called a Tusi-couple, which generates linear motion from the sum of two circular motions
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.
41.
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 as a civil servant in Nanyang. Eventually, he became Chief Astronomer, Prefect of the Majors for Official Carriages and his uncompromising stance on historical and calendrical issues led to his becoming a controversial figure, preventing him from rising to the status of Grand Historian. His political rivalry with the palace eunuchs during the reign of Emperor Shun led to his decision to retire from the court to serve as an administrator of Hejian in Hebei. Zhang returned home to Nanyang for a time, before being recalled to serve in the capital once more in 138. He died there a year later, in 139, Zhang applied his extensive knowledge of mechanics and gears in several of his inventions. He improved previous Chinese calculations for pi and his fu and shi poetry were renowned in his time and studied and analyzed by later Chinese writers. Zhang received many honors for his scholarship and ingenuity, some modern scholars have compared his work in astronomy to that of the Greco-Roman Ptolemy. Born in the town of Xie in Nanyang Commandery, Zhang Heng came from a distinguished, at age ten, Zhangs father died, leaving him in the care of his mother and grandmother. An accomplished writer in his youth, Zhang left home in the year 95 to pursue his studies in the capitals of Changan, while traveling to Luoyang, Zhang passed by a hot spring near Mount Li and dedicated one of his earliest fu poems to it. Government authorities offered Zhang appointments to offices, including a position as one of the Imperial Secretaries, yet he acted modestly. At age twenty-three, he returned home with the title Officer of Merit in Nanyang, serving as the master of documents under the administration of Governor Bao De, as he was charged with composing inscriptions and dirges for the governor, he gained experience in writing official documents. As Officer of Merit in the commandery, he was responsible for local appointments to office. He spent much of his time composing rhapsodies on the capital cities, when Bao De was recalled to the capital in 111 to serve as a minister of finance, Zhang continued his literary work at home in Xie. Zhang Heng began his studies in astronomy at the age of thirty and began publishing his works on astronomy, in 112, Zhang was summoned to the court of Emperor An, who had heard of his expertise in mathematics. When he was nominated to serve at the capital, Zhang was escorted by carriage—a symbol of his official status—to Luoyang and he was promoted to Chief Astronomer for the court, serving his first term from 115–120 under Emperor An and his second under the succeeding emperor from 126–132. As Chief Astronomer, Zhang was a subordinate of the Minister of Ceremonies, when the government official Dan Song proposed the Chinese calendar should be reformed in 123 to adopt certain apocryphal teachings, Zhang opposed the idea. He considered the teachings to be of questionable stature and believed they could introduce errors, others shared Zhangs opinion and the calendar was not altered, yet Zhangs proposal that apocryphal writings should be banned was rejected
Zhang Heng
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A stamp of Zhang Heng issued by China Post in 1955
Zhang Heng
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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
42.
Before Common Era
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Common Era or Current Era is a year-numbering system for the Julian and Gregorian calendars that refers to the years since the start of this era, i. e. since AD1. The preceding era is referred to as before the Common or Current Era, the Current Era notation system can be used as a secular alternative to the Dionysian era system, which distinguishes eras as AD and BC. The two notation systems are equivalent, thus 2017 CE corresponds to AD2017 and 400 BCE corresponds to 400 BC. The year-numbering system for the Gregorian calendar is the most widespread civil calendar used in the world today. For decades, it has been the standard, recognized by international institutions such as the United Nations. The expression has been traced back to Latin usage to 1615, as vulgaris aerae, the term Common Era can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish academics. He attempted to number years from a reference date, an event he referred to as the Incarnation of Jesus. Dionysius labeled the column of the table in which he introduced the new era as Anni Domini Nostri Jesu Christi, numbering years in this manner became more widespread in Europe with its usage by Bede in England in 731. Bede also introduced the practice of dating years before what he supposed was the year of birth of Jesus, in 1422, Portugal became the last Western European country to switch to the system begun by Dionysius. The first use of the Latin term vulgaris aerae discovered so far was in a 1615 book by Johannes Kepler, Kepler uses it again in a 1616 table of ephemerides, and again in 1617. A1635 English edition of that book has the title page in English – so far, a 1701 book edited by John LeClerc includes Before Christ according to the Vulgar Æra,6. A1716 book in English by Dean Humphrey Prideaux says, before the beginning of the vulgar æra, a 1796 book uses the term vulgar era of the nativity. The first so-far-discovered usage of Christian Era is as the Latin phrase aerae christianae on the page of a 1584 theology book. In 1649, the Latin phrase æræ Christianæ appeared in the title of an English almanac, a 1652 ephemeris is the first instance so-far-found for English usage of Christian Era. The English phrase common Era appears at least as early as 1708, a 1759 history book uses common æra in a generic sense, to refer to the common era of the Jews. The first-so-far found usage of the phrase before the era is in a 1770 work that also uses common era and vulgar era as synonyms. The 1797 edition of the Encyclopædia Britannica uses the terms vulgar era, the Catholic Encyclopedia in at least one article reports all three terms being commonly understood by the early 20th century. Thus, the era of the Jews, the common era of the Mahometans, common era of the world
Before Common Era
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Key concepts
43.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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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
44.
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 its endpoint is on a line, as a rotation, a right angle corresponds to a quarter turn. The presence of an angle in a triangle is the defining factor for right triangles. The term is a calque of Latin angulus rectus, here rectus means upright, 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+22BE ⊾ Right angle with arc, U+299C ⦜ Right angle variant with square, and U+299D ⦝ Measured right angle with dot. The symbol for an angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland. Right angles are fundamental in Euclids Elements and they are defined in Book 1, definition 10, which also defines perpendicular lines. Euclid uses right angles in definitions 11 and 12 to define acute angles, two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all angles are equal. Euclids commentator Proclus gave a proof of this using the previous postulates. Saccheri gave a proof as well but using a more explicit assumption, in Hilberts 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, 90° π/2 radians 100 grad 8 points 6 hours Throughout history carpenters and masons have known a quick way to confirm if an angle is a true right angle. It is based on the most widely known Pythagorean triple and so called the Rule of 3-4-5 and this measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem, Thales theorem states that an angle inscribed in a semicircle is a right angle. Two application examples in which the angle and the Thales theorem are included. Cartesian coordinate system Orthogonality Perpendicular Rectangle Types of angles Wentworth, G. A, Euclid, commentary and trans. by T. L. Heath Elements Vol.1 Google Books
Right angle
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A right angle is equal to 90 degrees.
45.
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 two sides. If the catheti of a triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor and major cathetus. In a right triangle, the length of a cathetus is the mean of the length of the adjacent segment cut by the altitude to the hypotenuse. 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, geographic Information Systems, An Introduction, 3rd ed. New York, Wiley, p.271,2002, Cathetus at Encyclopaedia of Mathematics Weisstein, Eric W. Cathetus
Cathetus
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A right-angled triangle where c 1 and c 2 are the catheti and h is the hypotenuse
46.
Theorem
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In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. 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 and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
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.
47.
Babylonian mathematics
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Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 gives an approximation to 2 accurate to three significant sexagesimal digits, Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, the Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute,60 minutes in an hour, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
Babylonian mathematics
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Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
48.
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 system, a binary system, algebra, geometry. Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, as in other early societies the focus was on astronomy in order to perfect the agricultural calendar, and other practical tasks, and not on establishing formal systems. Ancient Chinese mathematicians did not develop an approach, but made advances in algorithm development. 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 Pascals 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 developed a decimal system. Since early times, Chinese understood basic arithmetic, algebra, equations, although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry and the usage of decimals. Math was one of the Liù Yì or Six Arts, students were required to master during the Zhou Dynasty, learning them all perfectly was required to be a perfect gentleman, or in the Chinese sense, a Renaissance Man. Six Arts have their roots in the Confucian philosophy, the oldest existent work on geometry in China comes from the philosophical Mohist canon of c.330 BC, compiled by the followers of Mozi. The Mo Jing described various aspects of many associated with physical science. It provided a definition of the geometric point, stating that a line is separated into parts. Much like Euclids first and third definitions and Platos beginning of a line, there is nothing similar to it. Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit and it also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, the history of mathematical development lacks some evidence. There are still debates about certain mathematical classics, for example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300–250 BC
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
49.
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 law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclids Elements, dating back to the 3rd century BC, the cases of obtuse triangles and acute triangles are treated separately, in Propositions 12 and 13 of Book 2. Using notation as in Fig.2, Euclids statement can be represented by the formula A B2 = C A2 + C B2 +2 and this formula may be transformed into the law of cosines by noting that CH = cos = − cos γ. Proposition 13 contains an analogous statement for acute triangles. In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a 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, the theorem is used in triangulation, for solving a triangle or circle, i. e. These formulas produce high round-off errors in floating point calculations if the triangle is very acute and 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 a2 − 2ab cos γ + b2 − c2 =0 and this equation can have 2,1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ and these different cases are also explained by the side-side-angle congruence ambiguity. Consider a triangle with sides of length a, b, c and this triangle can be placed on the Cartesian coordinate system by plotting the following points, as shown in Fig.4, A =, B =, and 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. Drop the perpendicular onto the c to get c = a cos β + b cos α. Multiply through by c to get c 2 = a c cos β + b c cos α. By considering the other perpendiculars obtain a 2 = a c cos β + a b cos γ, b 2 = b c cos α + a b cos γ. Adding the latter two equations gives a 2 + b 2 = a c cos β + b c cos α +2 a b cos γ and this proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle
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.
50.
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 eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, 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 word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
Ratio
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The ratio of width to height of standard-definition television.
51.
Triangle postulate
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In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is equal to the straight angle, also expressed as 180°, π radians. It was unknown for a long time whether other geometries exist, the influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the answer was proven to be positive, in other spaces this sum can be greater or lesser and its difference from 180° 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, in the presence of the other axioms of Euclidean geometry, the following statements are equivalent, Triangle postulate, The sum of the angles of a triangle is two right angles. Playfairs axiom, Given a straight line and a point not on the line, 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, Three points property, Three points either 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 two sides. The sum of the angles of a triangle is less than 180°. The relation between angular defect and the area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Playfairs axiom, Proclus axiom, the equidistance postulate, a circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small, for an ideal triangle, a generalization of hyperbolic triangles, this sum is equal to zero. For a spherical triangle, the sum of the angles is greater than 180°, specifically, the sum of the angles is 180° ×, where f is the fraction of the spheres area which is enclosed by the triangle. Note that spherical geometry does not satisfy several of Euclids axioms Angles between adjacent sides of a triangle are referred to as interior angles in Euclidean and other geometries, exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the sum of all three angles, that equals to 360° in the Euclidean case, is less than 360° in the spherical case. Euclids Elements Foundations of geometry Hilberts axioms Saccheri quadrilateral Lambert quadrilateral
Triangle postulate
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Equivalence of the parallel postulate and the "sum of the angles equals to 180°" statement
52.
Parallel postulate
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In geometry, the parallel postulate, also called Euclids fifth postulate because it is the fifth postulate in Euclids Elements, is a distinctive axiom in Euclidean geometry. This postulate does not specifically talk about parallel lines, it is only a postulate related to parallelism, Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. Euclidean geometry is the study of geometry that satisfies all of Euclids axioms, a geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclids fifth postulate is known as absolute geometry and this axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the axioms which give Euclidean geometry, each of these can be used to prove the other. These equivalent statements include, There is at most one line that can be parallel to another given one through an external point. The sum of the angles in every triangle is 180°, There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle, There exists a pair of similar, but not congruent, triangles. If three angles of a quadrilateral are right angles, then the angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the line are also parallel to each other. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the two sides. There is no limit to the area of a triangle. The summit angles of the Saccheri quadrilateral are 90°, if a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. In the list above, it is taken to refer to non-intersecting lines. Note that the two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines. For two thousand years, many attempts were made to prove the parallel postulate using Euclids first four postulates, the main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found
Parallel postulate
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If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.
53.
U.S. Representative
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The United States House of Representatives is the lower chamber of the United States Congress which, along with the Senate, composes the legislature of the United States. 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, who is elected by the members thereof and is traditionally the leader of the controlling party. He or she and other leaders are chosen by the Democratic Caucus or the Republican Conferences. The House meets in the wing of the United States Capitol. Under the Articles of Confederation, the Congress of the Confederation was a body in which each state was equally represented. 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 house, with the Senate being the upper house. 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, and Pennsylvania, as it called for representation based on population. The smaller states, however, favored the New Jersey Plan, the Constitution was ratified by the requisite number of states in 1788, but its implementation was set for March 4,1789. The House began work on April 1,1789, when it achieved a quorum for the first time, during the first half of the 19th century, the House was frequently in conflict with the Senate over regionally divisive issues, including slavery. The North was much more populous than the South, and therefore dominated the House of Representatives, However, the North held no such advantage in the Senate, where the equal representation of states prevailed. Regional conflict was most pronounced over the issue of slavery, One example of a provision repeatedly supported by the House but blocked by the Senate was the Wilmot Proviso, which sought to ban slavery in the land gained during the Mexican–American War. Conflict over slavery and other issues persisted until the Civil War, the war culminated in the Souths defeat and in the abolition of slavery. Because all southern senators except Andrew Johnson resigned their seats at the beginning of the war, the years of Reconstruction that followed witnessed large majorities for the Republican Party, which many Americans associated with the Unions victory in the Civil War and the ending of slavery. The Reconstruction period ended in about 1877, the ensuing era, the Democratic and the Republican Party held majorities in the House at various times. The late 19th and early 20th centuries also saw an increase in the power of the Speaker of the House
U.S. Representative
–
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
–
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.
54.
Pythagorean triple
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A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and c are coprime, a right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
Pythagorean triple
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The Pythagorean theorem: a 2 + b 2 = c 2
55.
Coprime
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In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + 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 ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
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
56.
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 an isosceles right triangle, with each leg having unit length. The process then repeats, the i th triangle in the sequence is a triangle with side lengths √i and 1. For example, the 16th triangle has sides measuring 4,1 and hypotenuse of √17 Although all of Theodorus work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. Plato does not attribute the irrationality of the root of 2 to Theodorus. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories, each of the triangles hypotenuses hi gives the square root of the corresponding 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 unit length are extended into a line, they will never pass through any of the other vertices of the total figure. Theodorus stopped his spiral at the triangle 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, 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 angle φ for the kth triangle. It grows proportionally to the root of k, with a bounded correction term c2. The growth of the radius of the spiral at a certain triangle n is Δ r = n +1 − n, the Spiral of Theodorus approximates the Archimedean spiral. An alternative derivation is given in, some have suggested a different interpolant which connects the spiral and an alternative inner spiral, as in
Spiral of Theodorus
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The spiral of Theodorus up to the triangle with a hypotenuse of √17
57.
Compass and straightedge constructions
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in 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, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its 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 segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
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
58.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a 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.
59.
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 for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense 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 value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot 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.
60.
Complex plane
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In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the part of a complex number represented by a displacement along the x-axis. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is known as the Argand plane. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
Complex plane
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Geometric representation of z and its conjugate z̅ in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.
61.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and 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. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and 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 length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
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
62.
Polar coordinates
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The reference point is called the pole, and 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 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 century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—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 coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges 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, 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 Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
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
63.
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 that is locally invertible at each point and this means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space are Cartesian, cylindrical and spherical polar coordinates. A Cartesian coordinate surface in space is a coordinate plane. In the same space, the coordinate surface r =1 in spherical coordinates is the surface of a unit sphere. The formalism of curvilinear coordinates provides a unified and general description of the coordinate systems. Curvilinear coordinates are used to define the location or distribution of physical quantities which may be, for example, scalars, vectors. Such expressions then become valid for any curvilinear coordinate system, depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a 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 coordinate system may be easier to solve in that system. One would for instance describe the motion of a particle in a box in Cartesian coordinates. Spherical coordinates are one of the most used curvilinear coordinate systems in fields as Earth sciences, cartography, and physics. A point P in 3d space can be defined using Cartesian coordinates, by r = x e x + y e y + z e z and it can also be defined by its curvilinear coordinates if this triplet of numbers defines a single point in an unambiguous way. The coordinate axes are determined by the tangents to the curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates. Applying the same derivatives to the curvilinear system locally at point P defines the basis vectors. Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis, all bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, note, for this article e is reserved for the standard basis and h or b is for the curvilinear basis
Curvilinear coordinates
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Curvilinear, affine, and Cartesian coordinates in two-dimensional space
64.
Legendre polynomials
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In mathematics, Legendre functions are solutions to Legendres 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 Laplaces equation in spherical coordinates. The Legendre differential equation may be solved using the power series method. The equation has regular singular points at x = ±1 so, in general, when n is an integer, the solution Pn that is regular at x =1 is also regular at x = −1, and the series for this solution terminates. 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 and it may be expressed using Rodrigues formula, P n =12 n n. The Pn can also be defined as the coefficients in a Taylor series expansion, In physics, expanding the Taylor series in Equation for the first two terms gives P0 =1, P1 = x for the first two Legendre Polynomials. Replacing the quotient of the root with its definition in. 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. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution. Where z ^ is the axis of symmetry and θ is the angle between the position of the observer and the z ^ axis, the solution for the potential will be Φ = ∑ ℓ =0 ∞ P ℓ. A ℓ and B ℓ are to be determined according to the condition of each problem. They also appear when solving Schrödinger equation in three dimensions for a central force. Legendre polynomials are also useful in expanding functions of the form,11 + η2 −2 η x = ∑ k =0 ∞ η k P k which arise naturally in multipole expansions. The left-hand side of the equation is the function for the Legendre polynomials. This expansion is used to develop the normal multipole expansion, conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion, the trigonometric functions cos n θ, also denoted as the Chebyshev polynomials T n ≡ cos n θ, can also be multipole expanded by the Legendre polynomials P n
Legendre polynomials
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Figure 2
65.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
Trigonometry
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Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".
Trigonometry
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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
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Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
66.
Cross product
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In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. Given two linearly independent vectors a and b, the product, a × b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and it should not be confused with dot product. If two vectors have the direction or if either one has zero length, then their cross product is zero. The cross product is anticommutative and is distributive over addition, the space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, the cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used, if the vectors a and 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 = −. By pointing the forefinger toward b first, and then pointing the finger toward a. Using the cross product requires the handedness of the system to be taken into account. If a left-handed coordinate system is used, the direction of the n is given by the left-hand rule. This, however, creates a problem because transforming from one arbitrary reference system to another, the problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector. See cross product and handedness for more detail, in 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period and an x, respectively, to denote them. These alternative names are widely used in the literature. Both the cross notation and the cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b, as explained below, the cross product can be expressed in the form of a determinant of a special 3 ×3 matrix
Cross product
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The cross-product in respect to a right-handed coordinate system
67.
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. Sometimes it is called inner product in the context of Euclidean space, algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them, the dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and 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, so the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry. For instance, in space, the dot product of vectors and is. In Euclidean space, a Euclidean vector is an object that possesses both a magnitude and a direction. A vector can be pictured as an arrow and its magnitude is its length, and its direction is the direction that the arrow points. The magnitude of a vector a is denoted by ∥ a ∥, the dot product of two Euclidean vectors a and b is defined by a ⋅ b = ∥ a ∥ ∥ b ∥ cos , where θ is the angle between a and b. In particular, if a and b are orthogonal, then the angle between them is 90° and a ⋅ b =0. The scalar projection of a Euclidean vector a in the direction of a Euclidean vector b is given by a b = ∥ a ∥ cos θ, where θ is the angle between a and b. In terms of the definition of the dot product, this can be rewritten a b = a ⋅ b ^. The dot product is thus characterized geometrically by a ⋅ b = a b ∥ b ∥ = b a ∥ a ∥. The dot product, defined in this manner, is homogeneous under scaling in each variable and it also satisfies a distributive law, meaning that a ⋅ = a ⋅ b + a ⋅ c. These properties may be summarized by saying that the dot product is a bilinear form, moreover, this bilinear form is positive definite, which means that a ⋅ a is never negative and is zero if and only if a =0. En are the basis vectors in Rn, then we may write a = = ∑ i a i e i b = = ∑ i b i e i. The vectors ei are a basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length e i ⋅ e i =1 and since they form right angles with each other, thus in general we can say that, e i ⋅ e j = δ i j
Dot product
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Scalar projection
68.
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, the seven-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the cross product is unique up to a sign. The seven-dimensional cross product has the relationship to the octonions as the three-dimensional product does to the quaternions. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results, the product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of basis vectors ei, for example, from the table e 1 × e 2 = e 3 = − e 2 × e 1 The table can be used to calculate the product of any two vectors. This can be repeated for the six components. There are 480 such tables, 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, a third statement of the magnitude condition is | x × y | = | x | | y | if =0. Given the properties of bilinearity, orthogonality and magnitude, a cross product exists only in three and seven dimensions. This can be shown by postulating the properties required for the cross product, in zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both these cases the product must be identically zero. The restriction to 0,1,3 and 7 dimensions is related to Hurwitzs theorem, in contrast the three-dimensional cross product, which is unique, there are many possible binary cross products in seven dimensions. Unlike in three dimensions, x × y = a × b does not imply that a and b lie in the plane as x and y. One possible multiplication table is described in the Example section, unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is applied to general vectors x and y by expressing x and y in terms of the basis. More compactly this rule can be written as e i × e i +1 = e i +3 with i =1.7 modulo 7, together with anticommutativity this generates the product. This rule directly produces the two immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences, e i × = − e i +1 = e i × e i +3, which produces diagonals further out, and so on
Seven-dimensional cross product
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Fano planes for the two multiplication tables used here.
69.
Hippocrates of Chios
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Hippocrates of Chios was an ancient Greek mathematician, geometer, and 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 mathematician and astronomer Oenopides of Chios. The reductio ad absurdum argument has been traced to him, only a single, and famous, fragment of Hippocrates Elements is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some so-called Hippocratic lunes — see Lune of Hippocrates. This was part of a programme to achieve the quadrature of the circle. 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, the number π is the ratio of the circumference to the diameter of a circle, and also the ratio of the area to the square of the radius. In the century after Hippocrates at least four other mathematicians wrote their own Elements, steadily improving terminology, in this way Hippocrates pioneering work laid the foundation for Euclids Elements that was to remain the standard geometry textbook for many centuries. Hippocrates is believed to have originated the use of letters to refer to the points and figures in a proposition, e. g. triangle ABC for a triangle with vertices at points A, B. Two other contributions by Hippocrates in the field of mathematics are noteworthy and he found a way to tackle the problem of duplication of the cube, that is, the problem of how to construct a cube root. Like the quadrature of the circle this was another of the three great mathematical problems of Antiquity. Hippocrates also invented the technique of reduction, that is, to transform specific mathematical problems into a general problem that is more easy to solve. The solution to the general problem then automatically gives a solution to the original problem. In the field of astronomy Hippocrates tried to explain the phenomena of comets, ivor Bulmer-Thomas, Hippocrates of Chios, in, Dictionary of Scientific Biography, Charles Coulston Gillispie, ed. pp. 410–418. Björnbo, Hippokrates, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa, oConnor, John J. Robertson, Edmund F. Hippocrates of Chios, MacTutor History of Mathematics archive, University of St Andrews. The Quadrature of the Circle and Hippocrates Lunes at Convergence
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).
70.
Pappus of Alexandria
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Pappus of Alexandria was one of the last great Alexandrian mathematicians of Antiquity, known for his Synagoge or Collection, and for Pappuss hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, Collection, his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a range of topics, including geometry, recreational mathematics, doubling the cube, polygons. 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, in his surviving writings, Pappus gives no indication of the date of the authors whose works he makes use of, or of the time at which he himself wrote. If no other information were available, all that could be known would be that he was later than Ptolemy, whom he quotes, and earlier than Proclus. The Suda states that Pappus was of the age as Theon of Alexandria. A different date is given by a note to a late 10th-century manuscript, which states, next to an entry on Emperor Diocletian. This works out as October 18,320 AD, and so Pappus must have flourished c.320 AD. The great work of Pappus, in eight books and titled Synagoge or Collection, has not survived in complete form, the first book is lost, and the rest have suffered considerably. The Suda enumerates other works of Pappus, Χωρογραφία οἰκουμενική, commentary on the 4 books of Ptolemys Almagest, Ποταμοὺς τοὺς ἐν Λιβύῃ, Pappus himself mentions another commentary of his own on the Ἀνάλημμα of Diodorus of Alexandria. Pappus also wrote commentaries on Euclids Elements, and on Ptolemys Ἁρμονικά and these discoveries form, in fact, a text upon which Pappus enlarges discursively. Heath considered the systematic introductions to the books as valuable, for they set forth clearly an outline of the contents. From these introductions one can judge of the style of Pappuss writing, heath also found his characteristic exactness made his Collection a most admirable substitute for the texts of the many valuable treatises of earlier mathematicians of which time has deprived us. The portions of Collection which has survived can be summarized as follows and 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 a book by Apollonius of Perga. The final propositions deal with multiplying together the values of Greek letters in two lines of poetry, producing two very large numbers approximately equal to 2*1054 and 2*1038. Book III contains geometrical problems, plane and solid, on the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure
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
71.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
72.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert 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 space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist
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.
73.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and 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 to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
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.
74.
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. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, 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 to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
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.
75.
Orthogonality
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, and engineering. The word comes from the Greek ὀρθός, meaning upright, and γωνία, the ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle, in the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i. e. they form a right angle, two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y, two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a subspace is its orthogonal complement. Given a module M and its dual M∗, an element m′ of M∗, two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent, a set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set, nonzero pairwise orthogonal vectors are always linearly independent. In certain cases, the normal is used to mean orthogonal. For example, the y-axis is normal to the curve y = x2 at the origin, however, normal may also refer to the magnitude of a vector. In particular, a set is called if it is an orthogonal set of unit vectors. As a result, use of the normal to mean orthogonal is often avoided. The word normal also has a different meaning in probability and statistics, a vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality, in the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ. In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i. e. they make an angle of 90°, hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces
Orthogonality
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The line segments AB and CD are orthogonal to each other.
76.
Spherical law of cosines
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In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a triangle on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere. Since this is a sphere, the lengths a, b. As a special case, for C = π/2, then cos C =0, 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 formulation of the law of haversines is preferable. It can be obtained from consideration of a spherical triangle dual to the given one, a proof of the law of cosines can be constructed as follows. Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. Then, the angle C is given by, cos C = t a ⋅ t b = cos c − cos a cos b sin a sin b from which the law of cosines immediately follows. To the diagram above, add a plane tangent to the sphere at u and 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. Sides of the first triangle are tan a and tan b, so − tan a tan b cos C =1 − sec a sec b cos c Multiply both sides by cos a cos b and rearrange. The angles and distances do not change if the sphere is rotated, so we can rotate the sphere so that u is at the north pole, with this rotation, the spherical coordinates for v are = and the spherical coordinates for w are =. The Cartesian coordinates for v are = and the Cartesian coordinates for w are =
Spherical law of cosines
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Spherical triangle solved by the law of cosines.
77.
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, the way to indicate this and represent the answer to 10 significant figures is, 6990100000000000000♠1. Furthermore, it usually only postpones the problem, What if the data is accurate to ten digits. One of the most important parts of analysis is to avoid or minimize loss of significance in calculations. If the underlying problem is well-posed, there should be an algorithm for solving it. Let x and y be positive normalized floating point numbers, in the subtraction x − y, r significant bits are lost where q ≤ r ≤ p 2 − p ≤1 − y x ≤2 − q for some positive integers p and q. For example, consider the equation, a x 2 + b x + c =0. 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. The case a =1, b =200, c = −0.000015 will serve to illustrate the problem, x 2 +200 x −0.000015 =0. We have b 2 −4 a c =2002 +4 ×1 ×0.000015 =200.00000015 … In real arithmetic, in 10-digit floating-point arithmetic, /2 = −200.00000005, /2 =0.00000005. Notice that the solution of greater magnitude is accurate to ten digits, because of the subtraction that occurs in the quadratic equation, it does not constitute a stable algorithm to calculate the two roots. A careful floating point computer implementation combines several strategies to produce a robust result, here sgn denotes the sign function, where sgn is 1 if b is positive and −1 if b is negative. This avoids cancellation problems between b and the root of the discriminant by ensuring that only numbers of the same sign are added. To illustrate the instability of the quadratic formula versus this variant formula. The discriminant b 2 −4 a c needs to be computed in arithmetic of twice the precision of the result to avoid this and this can be in the form of a fused multiply-add operation
Loss of significance
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Example of LOS in case of computing 2 forms of the same function
78.
Gaussian curvature
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In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point, K = κ1 κ2. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane, the Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is a measure of curvature, depending only on distances that are measured on the surface. This is the content of the Theorema egregium, Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827. At any point on a surface, we can find a vector that is at right angles to the surface. The intersection of a plane and the surface will form a curve called a normal section. For most points on most surfaces, different normal sections will have different curvatures, 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, if both principal curvatures are of the same sign, κ1κ2 >0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. 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. If the principal curvatures have different signs, κ1κ2 <0, then the Gaussian curvature is negative, at such points, the surface will be saddle shaped. If one of the principal curvatures is zero, κ1κ2 =0, the Gaussian curvature is zero, most surfaces will contain regions of positive Gaussian curvature and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line. When a surface has a constant zero Gaussian curvature, then it is a developable surface, when a surface has a constant positive Gaussian curvature, then it is a sphere and the geometry of the surface is spherical geometry. When a surface has a constant negative Gaussian curvature, then it is a pseudospherical surface, in differential 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 by different amounts in different directions at that point. We represent the surface by the implicit function theorem as the graph of a function, f, then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point and it is also given by K = ⟨ e 1, e 2 ⟩ det g, where ∇ i = ∇ e i is the covariant derivative and g is the metric tensor. At a point p on a surface in R3, the Gaussian curvature is also given by K = det. A useful formula for the Gaussian curvature is Liouvilles equation in terms of the Laplacian in isothermal coordinates, the surface integral of the Gaussian curvature over some region of a surface is called the total curvature
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).
79.
Hyperbolic function
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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular functions. The inverse hyperbolic functions are the hyperbolic sine arsinh and so on. Just as the form a circle with a unit radius. 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 triangle covering this sector. Laplaces equations are important in areas of physics, including electromagnetic theory, heat transfer, fluid dynamics. In complex analysis, the hyperbolic functions arise as the parts of sine and cosine. When considered defined by a variable, the hyperbolic functions are rational functions of exponentials. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati, 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 hyperbolic functions are, Hyperbolic sine, sinh x = e x − e − x 2 = e 2 x −12 e x =1 − e −2 x 2 e − x. Hyperbolic cosine, cosh x = e x + e − x 2 = e 2 x +12 e x =1 + e −2 x 2 e − x, the complex forms in the definitions above derive from Eulers formula. One also has sech 2 x =1 − tanh 2 x csch 2 x = coth 2 x −1 for the other functions, sinh = sinh 2 = sgn cosh −12 where sgn is the sign function. All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x and e − x, and it is possible to express the above functions as Taylor series, sinh x = x + x 33. + ⋯ = ∑ n =0 ∞ x 2 n +1, the function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, the function cosh x has a Taylor series expression with only even exponents for x. Thus it is a function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the series expression of the exponential 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
80.
Plimpton 322
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Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G. A and this tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists what are now called Pythagorean triples, i. e. integers a, b, from a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. Although the tablet was interpreted in the past as a table, more recently published work sees this as anachronistic. For readable popular treatments of this tablet see Robson or, more briefly, Robson is a more detailed and technical discussion of the interpretation of the tablets numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide,9 cm tall, according to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa. More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Robson points out that Plimpton 322 was written in the same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, the fourth column is just a row number, in order from 1 to 15. The second and third columns are visible in the surviving tablet. Conversion of these numbers from sexagesimal to decimal raises additional ambiguities, the sixty sexigesimal entries are exact, no truncations or rounding off. In each row, the number in the column can be interpreted as the shortest side s of a right triangle. The number in the first column is either the fraction s 2 l 2 or d 2 l 2 =1 + s 2 l 2, scholars still differ, however, on how these numbers were generated. Below is the translation of the tablet. Otto E. Neugebauer argued for an interpretation, pointing out that this table provides a list of Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side, hypotenuse ratio of the familiar right triangle. If p and q are two numbers, one odd and one even, then form a Pythagorean triple. For instance, line 11 can be generated by this formula with p =2 and q =1, as Neugebauer argues, each line of the tablet can be generated by a pair that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, neugebauers explanation is the one followed e. g. by Conway & Guy
Plimpton 322
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The Plimpton 322 tablet.
81.
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 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 surrounding regions and written records. Not much is known about the kings from Sumuabum 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 consisted of a few towns in the surrounding area, Dilbat, Sippar, Kish. Once Hammurabi was king, his military victories gained land for the empire, however, Babylon remained but one of several important areas in Mesopotamia, along with Assyria, then ruled by Shamshi-Adad I, and Larsa, then ruled by Rim-Sin I. In Hammurabis thirtieth year as king, he 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. In essence, Hammurabi gained control over all of south Mesopotamia, the other formidable political power in the region in the 2nd millennium was Eshnunna, which Hammurabi succeeded in capturing in c. Babylon exploited Eshnunnas well-established commercial trade routes and the stability that came with them. It was not long before Hammurabis army took Assyria and parts of the Zagros Mountains, Hammurabis other name was Hammurapi-ilu, meaning Hammurapi the god or perhaps Hammurapi is god. He could have been Amraphel king of Shinar or Sinear in the Jewish records and the Bible, Abraham lived from 1871 to 1784, according to modern interpretations of the Old Testaments figures that have been usually reckoned in modern half years before the Exodus, from equinox to equinox. The Venus tablets of Ammisaduqa are famous, and several books had been published about them, several dates have been offered but the old dates of many sourcebooks seems to be outdated and incorrect. A few sources, some printed almost a century ago, claim that the text mentions an occultation of the Venus by the moon. However, this may be a misinterpretation, calculations support 1659 for the fall of Babylon, based on the statistical probability of dating based on the planets observations. The presently accepted middle chronology is too low from the point of view. A text about the fall of Babylon by the Hittites of Mursilis I at the end of Samsuditanas reign which tells about an eclipse is crucial for a correct Babylonian chronology. The pair of lunar and solar eclipses occurred in the month Shimanu, the lunar eclipse took place on February 9,1659 BC
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
82.
Stigler's law of eponymy
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Stiglers 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 Stiglers law to show that it follows its own decree. In the case of eponymy, the idea becomes named after that person, 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 he added his little mite — that is all he did. These object lessons should teach us that ninety-nine parts of all things that proceed from the intellect are plagiarisms, pure and simple, Stephen Stiglers 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, the Matthew effect was coined by Robert K. The effect applies specifically to women through the Matilda effect, boyers law was named by Hubert Kennedy in 1972. Kennedy observed that it is 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, and Other Essays. Chicago, The University of Chicago Press, Stigler, Stephen M. Gieryn, F. ed. Stiglers law of eponymy. Transactions of the New York Academy of Sciences, ciphering, Sectorials, D Lesions, Freckles and the Operation of Stiglers Law. Eponymy and Laws of Eponymy. on Miller, Jeff, earliest known uses of some of the words of mathematics. In the Air, Who says big ideas are rare, Stiglers law is described near the end of the article
Stigler's law of eponymy
83.
Leon Lederman
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He is Director Emeritus of Fermi National Accelerator Laboratory in Batavia, Illinois, USA. He founded the Illinois Mathematics and Science Academy, in Aurora, Illinois in 1986, in 2012, he was awarded the Vannevar Bush Award for his extraordinary contributions to understanding the basic forces and particles of nature. Lederman was born in New York City, New York, the son of Minna and Morris Lederman, Lederman graduated from the James Monroe High School in the South Bronx. He received his bachelors degree from the City College of New York in 1943 and he then joined the Columbia faculty and eventually became Eugene Higgins Professor of Physics. In 1960, on leave from Columbia, he spent some time at CERN in Geneva as a Ford Foundation Fellow and 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. Also known as Right-side Up Science and Biology Last, this movement seeks to rearrange the current high school curriculum so that physics precedes chemistry. A former president of the American Physical Society, Lederman also received the National Medal of Science, the Wolf Prize, Lederman served as President of the Board of Sponsors of The Bulletin of the Atomic Scientists. He also served on the board of trustees for Science Service, now known as Society for Science & the Public, from 1989 to 1992, 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, in 1977, a group of physicists, the E288 experiment team, led by Leon Lederman announced that a particle with a mass of about 6.0 GeV was being produced by the Fermilab particle accelerator. The particles initial name was the greek letter Upsilon, after taking further data, the group discovered that this particle did not actually exist, and the discovery was named Oops-Leon as a pun on the original name and Ledermans first name. Lederman later wrote his 1993 popular science book The God Particle, If the Universe Is the Answer, – which sought to promote awareness of the significance of such a project – in the context of the projects last years and the changing political climate of the 1990s. The increasingly moribund project was finally shelved that same year after some $2 billion of expenditures, Lederman also received the National Medal of Science, the Elliott Cresson Medal for Physics, the Wolf Prize for Physics and the Enrico Fermi Award. In 1995, he received the Chicago History Museum Making History Award for Distinction in Science Medicine, Lederman was an early supporter of Science Debate 2008, an initiative to get the then-candidates for president, Barack Obama and John McCain, to debate the nations top science policy challenges. Lederman was also a member of the USA Science and Engineering Festivals Advisory Board, Lederman was born in New York to a family of Jewish immigrants from Russia. His father operated a hand laundry while encouraging Leon to pursue his education and he went to elementary school in New York City, continuing on to college and his doctorate in the city. In his book, The God Particle, If the Universe Is the Answer, Lederman wrote that, although he was a chemistry major, he became fascinated with physics, because of the clarity of the logic and the unambiguous results from experimentation. His best friend during his years, Martin Klein, convinced him of the splendors of physics during a long evening over many beers
Leon Lederman
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Lederman on May 11, 2007
84.
Egypt
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Egypt, officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia by a land bridge formed by the Sinai Peninsula. Egypt is a Mediterranean country bordered by the Gaza Strip and Israel to the northeast, the Gulf of Aqaba to the east, the Red Sea to the east and south, Sudan to the south, and Libya to the west. Across the Gulf of Aqaba lies Jordan, and across from the Sinai Peninsula lies Saudi Arabia, although Jordan and it is the worlds only contiguous Afrasian nation. Egypt has among the longest histories of any country, emerging as one of the worlds 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 religion and central government. One of the earliest centres of Christianity, Egypt was Islamised in the century and remains a predominantly Muslim country. With over 92 million inhabitants, Egypt is the most populous country in North Africa and the Arab world, the third-most populous in Africa, and the fifteenth-most populous in the world. The great majority of its people live near the banks of the Nile River, an area of about 40,000 square kilometres, the large regions of the Sahara desert, which constitute most of Egypts territory, are sparsely inhabited. About half of Egypts residents live in areas, with most spread across the densely populated centres of greater Cairo, Alexandria. Modern Egypt is considered to be a regional and middle power, with significant cultural, political, and military influence in North Africa, the Middle East and the Muslim world. Egypts economy is one of the largest and most diversified in the Middle East, Egypt is a member of the United Nations, Non-Aligned Movement, Arab League, African Union, and Organisation of Islamic Cooperation. Miṣr is the Classical Quranic Arabic and modern name of Egypt. The name is of Semitic origin, directly cognate with other Semitic words for Egypt such as the Hebrew מִצְרַיִם, the oldest attestation of this name for Egypt is the Akkadian
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.
85.
Berlin Papyrus 6619
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The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is an ancient Egyptian papyrus document from the Middle Kingdom, second half of the 12th or 13th dynasty. The two readable fragments were published by Hans Schack-Schackenburg in 1900 and 1902, the papyrus is one of the primary sources of ancient 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 ½ + ¼ the side of the other. The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x2 + y2 =100 and x = y reduce to the equation in y,2 + y2 =100. 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
86.
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 and he extended Babylons 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, Babylon was one of the many largely Amorite ruled city-states that dotted the central and southern Mesopotamian plains and waged war on each other for control of fertile agricultural land. 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, 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 Hammurabis reign were quite peaceful, Hammurabi used his power to undertake a series of public works, including heightening the city walls for defensive purposes, and expanding the temples. In c.1801 BC, the kingdom of Elam. With allies among the states, Elam attacked and destroyed the kingdom of Eshnunna, destroying a number of cities. In order to consolidate its position, Elam tried to start a war between Hammurabis Babylonian kingdom and the kingdom of Larsa. Hammurabi and the king of Larsa made an alliance when they discovered this duplicity and were able to crush the Elamites, although Larsa did not contribute greatly to the military effort. Angered by Larsas failure to come to his aid, Hammurabi turned on that southern power, as Hammurabi was assisted during the war in the south by his allies from the north such as Yamhad and Mari, the absence of soldiers in the north led to unrest. Continuing his expansion, Hammurabi turned his attention northward, quelling the unrest, next the Babylonian armies conquered the remaining northern states, including Babylons former ally Mari, although it is possible that the conquest of Mari was a surrender without any actual conflict. Hammurabi entered into a war with Ishme-Dagan I of Assyria for control of Mesopotamia. Eventually Hammurabi prevailed, ousting Ishme-Dagan I just before his own death, mut-Ashkur the new king of Assyria was forced to pay tribute to Hammurabi, however Babylon did not rule Assyria directly. In just a few years, Hammurabi had succeeded in uniting all of Mesopotamia under his rule, however, one stele of Hammurabi has been found as far north as Diyarbekir, where he claims the title King of the Amorites. Vast numbers of contract tablets, dated to the reigns of Hammurabi and his successors, have been discovered, as well as 55 of his own letters
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
87.
Isosceles
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. 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 legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and 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 that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
Isosceles
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Isosceles triangle with vertical axis of symmetry
88.
China
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China, officially the Peoples Republic of China, is a unitary sovereign state in East Asia and the worlds most populous country, with a population of over 1.381 billion. The state is governed by the Communist Party of China and its capital is Beijing, the countrys major urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin and Hong Kong. China is a power and a major regional power within Asia. Chinas landscape is vast and diverse, ranging from forest steppes, the Himalaya, Karakoram, Pamir and Tian Shan mountain ranges separate China from much of South and Central Asia. The Yangtze and Yellow Rivers, the third and sixth longest in the world, respectively, Chinas coastline along the Pacific Ocean is 14,500 kilometers long and is bounded by the Bohai, Yellow, East China and South China seas. China emerged as one of the worlds earliest civilizations in the basin of the Yellow River in the North China Plain. For millennia, Chinas political system was based on hereditary monarchies known as dynasties, in 1912, the Republic of China replaced the last dynasty and ruled the Chinese mainland until 1949, when it was defeated by the communist Peoples Liberation Army in the Chinese Civil War. The Communist Party established the Peoples Republic of China in Beijing on 1 October 1949, both the ROC and PRC continue to claim to be the legitimate government of all China, though the latter has more recognition in the world and controls more territory. China had the largest economy in the world for much of the last two years, during which it has seen cycles of prosperity and decline. Since the introduction of reforms in 1978, China has become one of the worlds fastest-growing major economies. As of 2016, it is the worlds second-largest economy by nominal GDP, China is also the worlds largest exporter and second-largest importer of goods. China is a nuclear weapons state and has the worlds largest standing army. The PRC is a member of the United Nations, as it replaced the ROC as a permanent member of the U. N. Security Council in 1971. China is also a member of numerous formal and informal multilateral organizations, including the WTO, APEC, BRICS, the Shanghai Cooperation Organization, the BCIM, the English name China is first attested in Richard Edens 1555 translation of the 1516 journal of the Portuguese explorer Duarte Barbosa. The demonym, that is, the name for the people, Portuguese China is thought to derive from Persian Chīn, and perhaps ultimately from Sanskrit Cīna. Cīna was first used in early Hindu scripture, including the Mahābhārata, there are, however, other suggestions for the derivation of China. The official name of the state is the Peoples Republic of China. The shorter form is China Zhōngguó, from zhōng and guó and it was then applied to the area around Luoyi during the Eastern Zhou and then to Chinas Central Plain before being used as an occasional synonym for the state under the Qing
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Yinxu, ruins of an ancient palace dating from the Shang Dynasty (14th century BCE)
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Some of the thousands of life-size Terracotta Warriors of the Qin Dynasty, c. 210 BCE
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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.
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The Nine Chapters on the Mathematical Art
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The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving texts from China. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution, and these were commented on by Liu Hui in the 3rd century. The method of chapter 7 was not found in Europe until the 13th 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 its influence on mathematical thought in China persisted until the Qing Dynasty era. Liu Hui wrote a detailed commentary on this book in 263. Lius commentary is of great mathematical interest in its own right, the Nine Chapters is an anonymous work, and its origins are not clear. This is no longer the case, the Suàn shù shū or writings on reckoning is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have closed in 186 BCE. While its relationship to the Nine Chapters is still under discussion by scholars, the Zhoubi Suanjing, a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong. Contents of The Nine Chapters are as follows, 方田 Fangtian - Bounding fields, areas of fields of various shapes, manipulation of vulgar fractions. Liu Huis commentary includes a method for calculation of π and the value of 3.14159. 粟米 Sumi - Millet and rice, exchange of commodities at different rates, pricing. Distribution of commodities and money at proportional rates, deriving arithmetic and geometric sums, division by mixed numbers, extraction of square and cube roots, diameter of sphere, perimeter and diameter of circle. 商功 Shanggong - Figuring for construction, volumes of solids of various shapes. 盈不足 Yingbuzu - Excess and deficit, linear problems solved using the principle known later in the West as the rule of false position
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A page of The Nine Chapters on the Mathematical Art
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History
90.
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, Plutarchs surviving works were written in Greek, but intended for both Greek and Roman readers. Plutarch was born to a prominent family in the town of Chaeronea, about 80 km east of Delphi. The name of Plutarchs father has not been preserved, but based on the common Greek custom of repeating a name in alternate generations, the name of Plutarchs grandfather was Lamprias, as he attested in Moralia and in his Life of Antony. His brothers, Timon and Lamprias, are mentioned in his essays and dialogues. Rualdus, in his 1624 work Life of Plutarchus, recovered the name of Plutarchs wife, Timoxena, from internal evidence afforded by his writings. A letter is still extant, addressed by Plutarch to his wife, bidding her not to grieve too much at the death of their two-year-old daughter, interestingly, he hinted at a belief in reincarnation in that letter of consolation. The exact number of his sons is not certain, although two of them, Autobulus and the second Plutarch, are often mentioned. Plutarchs treatise De animae procreatione in Timaeo is dedicated to them, another person, Soklarus, is spoken of in terms which seem to imply that he was Plutarchs son, but this is nowhere definitely stated. Plutarch studied mathematics and philosophy at the Academy of Athens under Ammonius from 66 to 67, at some point, Plutarch took Roman citizenship. He lived most of his life at Chaeronea, and was initiated into the mysteries of the Greek god Apollo. For many years Plutarch served as one of the two priests at the temple of Apollo at Delphi, the site of the famous Delphic Oracle, twenty miles from his home. By his writings and lectures Plutarch became a celebrity in the Roman Empire, yet he continued to reside where he was born, at his country estate, guests from all over the empire congregated for serious conversation, presided over by Plutarch in his marble chair. Many of these dialogues were recorded and published, and the 78 essays, Plutarch held the office of archon in his native municipality, probably only an annual one which he likely served more than once. He busied himself with all the matters of the town. The Suda, a medieval Greek encyclopedia, states that Emperor Trajan made Plutarch procurator of Illyria, however, most historians consider this unlikely, since Illyria was not a procuratorial province, and Plutarch probably did not speak Illyrian. Plutarch spent the last thirty years of his serving as a priest in Delphi. He thus connected part of his work with the sanctuary of Apollo, the processes of oracle-giving
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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.
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Parallel Lives, Amyot translation, 1565
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Plutarch's bust at Chaeronea, his home town.
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A page from the 1470 Ulrich Han printing of Plutarch's Parallel Lives.
91.
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 the Brief Lives and he was a pioneer archaeologist, who recorded numerous megalithic and other field monuments in southern England, and who is particularly noted as the discoverer of the Avebury henge monument. The Aubrey holes at Stonehenge are named after him, although there is doubt as to whether the holes that he observed are those that currently bear the name. He was also a pioneer folklorist, collecting together a miscellany of material on customs, traditions and 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 and he had wider interests in applied mathematics and astronomy, and was friendly with many of the greatest scientists of the day. For much of the 19th and 20th centuries, thanks largely to the popularity of Brief Lives, Aubrey was regarded as more than an entertaining but quirky, eccentric. Only in the 1970s did the full breadth and innovation of his begin to be more widely appreciated. He published little in his lifetime, and many of his most important manuscripts remain unpublished, or published only in partial and unsatisfactory form. Aubrey was born at Easton Piers or Percy, near Kington St Michael, Wiltshire, to a long-established and his grandfather, Isaac Lyte, lived at Lytes Cary Manor, Somerset, now owned by the National Trust. Richard Aubrey, his father, owned lands in Wiltshire and Herefordshire, for many years 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 read such books as came his way, including Bacons Essays, and studied geometry in secret. He was educated at the Malmesbury grammar school under Robert Latimer and he then studied at the grammar school at Blandford Forum, Dorset. He entered Trinity College, Oxford, in 1642, but his studies were interrupted by the English Civil War and his earliest antiquarian work dates from this period in Oxford. In 1646 he became a student of the Middle Temple and he spent a pleasant time at Trinity in 1647, making friends among his Oxford contemporaries, and collecting books. He was to show Avebury to Charles II at the Kings request in 1663 and his father died in 1652, leaving Aubrey large estates, but with them some complicated debts. He claimed that his memory was not tenacious by 17th-century standards, but from the early 1640s he kept notes of observations in natural philosophy, his friends ideas. He also began to write Lives of scientists in the 1650s, in 1659 he was recruited to contribute to a collaborative county history of Wiltshire, leading to his unfinished collections on the antiquities and the natural history of the county. His erstwhile friend and fellow-antiquary Anthony Wood predicted that he would one day break his neck while running downstairs in haste to interview some retreating guest or other and he drank the Kings health in Interregnum Herefordshire, but with equal enthusiasm attended meetings in London of the republican Rota Club
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Part of the southern inner ring at Avebury
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An early photograph of Stonehenge taken July 1877
92.
Hans Christian Andersen
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Hans Christian Andersen (/ˈhɑːnz ˈkrɪstʃən ˈændərsən/, Danish, often referred to in Scandinavia as H. C. Although a prolific writer of plays, travelogues, novels, and poems, Andersens popularity is not limited to children, his stories, called eventyr in Danish, express themes that transcend age and nationality. Some of his most famous fairy tales include The Emperors New Clothes, The Little Mermaid, The Nightingale, The Snow Queen, The Ugly Duckling, Thumbelina and his stories have inspired ballets, animated and live-action films and plays. Hans Christian Andersen was born in the town of Odense, Denmark, Andersens father, also Hans, considered himself related to nobility. His paternal grandmother had told his father that their family had in the past belonged to a social class. A persistent theory suggests that Andersen was a son of King Christian VIII. Andersens father, who had received an education, introduced Andersen to literature. Andersens mother, Anne Marie Andersdatter, was uneducated and worked as a washerwoman following his fathers death in 1816, she remarried in 1818. Andersen was sent to a school for poor children where he received a basic education and was forced to support himself, working as an apprentice for a weaver and, later. At 14, he moved to Copenhagen to seek employment as an actor, having an excellent soprano voice, he was accepted into the Royal Danish Theatre, but 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. Jonas Collin, director of the Royal Danish Theatre, felt a great affection for Andersen and sent him to a school in Slagelse. Andersen had already published his first story, The Ghost at Palnatokes Grave, though not a keen pupil, he also attended school at Elsinore until 1827. He later said his years in school were the darkest and most bitter of his life, at one school, he lived at his schoolmasters home. There he was abused and was told that it was to improve his character and he later said the faculty had discouraged him from writing in general, causing him to enter a state of depression. A very early fairy tale by Andersen, called The Tallow Candle, was discovered in a Danish archive in October 2012, the story, written in the 1820s, was about a candle who did not feel appreciated. It was written while Andersen was still in school and dedicated to a benefactor, in 1829, Andersen enjoyed considerable success with the short story A Journey on Foot from Holmens Canal to the East Point of Amager. Its protagonist meets characters ranging from Saint Peter to a talking cat, Andersen followed this success with a theatrical piece, Love on St. Nicholas Church Tower, and a short volume of poems
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Photograph taken by Thora Hallager, 1869
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Andersen's childhood home in Odense
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Paper chimney sweep cut by Andersen
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Painting of Andersen, 1836, by Christian Albrecht Jensen
93.
Gilbert and Sullivan
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Gilbert and Sullivan refers to the Victorian-era theatrical partnership of the librettist W. S. Gilbert and the composer Arthur Sullivan and to the works they jointly created. 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. Sullivan, six years Gilberts junior, composed the music, contributing memorable melodies that could convey both humour and pathos and their operas have enjoyed broad and enduring international success and are still performed frequently throughout the English-speaking world. Gilbert and Sullivan introduced innovations in content and form that influenced the development of musical theatre through the 20th century. The operas have influenced political discourse, literature, film. Producer Richard DOyly Carte brought Gilbert and Sullivan together and nurtured their collaboration and he built the Savoy Theatre in 1881 to present their joint works and founded the DOyly Carte Opera Company, which performed and promoted Gilbert and Sullivans works for over a century. Gilbert was born in London on 18 November 1836 and his father, William, was a naval surgeon who later wrote novels and short stories, some of which included illustrations by his son. Director and playwright Mike Leigh described the Gilbertian style as follows, With great fluidity and freedom, First, within the framework of the story, he makes bizarre things happen, and turns the world on its head. Thus the Learned Judge marries the Plaintiff, the soldiers metamorphose into aesthetes, and so on and his genius is to fuse opposites with an imperceptible sleight of hand, to blend the surreal with the real, and the caricature with the natural. In other words, to tell a perfectly outrageous story in a deadpan way. Gilbert developed his theories on the art of stage direction. At the time Gilbert began writing, theatre in Britain was in disrepute, Gilbert helped to reform and elevate the respectability of the theatre, especially beginning with his six short family-friendly comic operas, or entertainments, for Thomas German Reed. At a rehearsal for one of these entertainments, Ages Ago, the composer Frederic Clay introduced Gilbert to his friend, two years later, Gilbert and Sullivan would write their first work together. Those two intervening years continued to shape Gilberts theatrical style, Sullivan was born in London on 13 May 1842. His father was a bandmaster, and by the time Arthur had reached the age of eight. In school he began to compose anthems and songs, in 1856, he received the first Mendelssohn Scholarship and studied at the Royal Academy of Music and then at Leipzig, where he also took up conducting. His graduation piece, completed in 1861, was a suite of music to Shakespeares The Tempest. Revised and expanded, it was performed at the Crystal Palace in 1862 and was an immediate sensation and he began building a reputation as Englands most promising young composer, composing a symphony, a concerto, and several overtures, among them the Overture di Ballo, in 1870
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W. S. Gilbert
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Arthur Sullivan
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One of Gilbert's illustrations for his Bab Ballad "Gentle Alice Brown"
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An early poster showing scenes from The Sorcerer, Pinafore, and Trial by Jury
94.
The Pirates of Penzance
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The Pirates of Penzance, or, The Slave of Duty is a comic opera in two acts, with music by Arthur Sullivan and libretto by W. S. Gilbert. The operas official premiere was at the Fifth Avenue Theatre in New York City on 31 December 1879, where the show was well received by both audiences and critics. Its London debut was on 3 April 1880, at the Opera Comique, the story concerns Frederic, who, having completed his 21st year, is released from his apprenticeship to a band of tender-hearted pirates. He meets Mabel, the daughter of Major-General Stanley, and the two people fall instantly in love. Frederic soon learns, however, that he was born on the 29th of February and 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, Frederics only solace is that Mabel agrees to wait for him faithfully, Pirates was the fifth Gilbert and Sullivan collaboration and introduced the much-parodied Major-Generals Song. The opera was performed for over a century by the DOyly Carte Opera Company in Britain and by other opera companies. Pirates remains popular today, taking its place along with The Mikado, 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 premiere in the United States. At the time, American law offered no protection to foreigners. After the pairs previous opera, H. M. S, however, Gilbert, Sullivan, and their producer, Richard DOyly Carte, failed in their efforts, over the next decade, to control the American performance copyrights to Pirates and their other operas. Fiction and plays about pirates were ubiquitous in the 19th century, walter Scotts The Pirate and James Fenimore Coopers 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, Sullivan had written a comic opera called The Contrabandista, in 1867, about a hapless British tourist who is captured by bandits and forced to become their chief. Gilbert had written several works that involved pirates or bandits. In Gilberts 1876 opera Princess Toto, the character is eager to be captured by a brigand chief. Gilbert had translated Jacques Offenbachs operetta Les brigands, in 1871, as in Les brigands, The Pirates of Penzance absurdly treats stealing as a professional career path, with apprentices and tools of the trade such as the crowbar and life preserver. While Pinafore was running strongly at the Opera Comique in London, Gilbert was eager to get started on his and Sullivans next opera and he re-used several elements of his 1870 one-act piece, Our Island Home, which had introduced a pirate chief, Captain Bang. Bang was mistakenly apprenticed to a band as a child by his deaf nursemaid
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Drawing of the Act I finale
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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.
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George Grossmith as General Stanley, wearing Wolseley 's trademark moustache
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Catherine Ferguson (Kate), Nellie Briercliffe (Edith), and Ella Milne (Isabel), 1920
95.
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 a play and several movies, most famously the classic 1939 movie. 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. Believing he is the man capable of solving their problems, Dorothy and her friends travel to the Emerald City. Oz is very reluctant to meet them, but eventually each is granted an audience, one by one. In each of these occasions, the Wizard appears in a different form, once as a giant head, once as a fairy, once as a ball of fire. When, at last, he grants an audience to all of them at once, 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, as Oz had no leadership at the time, he became Supreme Ruler of the kingdom, and did his best to sustain the myth. He leaves Oz at the end of the novel, again in a hot air balloon, after the Wizards departure, the Scarecrow is briefly enthroned, until Princess Ozma is freed from the witch Mombi at the end of The Marvelous Land of Oz. In The Marvelous Land of Oz, the Wizard is described as having usurped the throne of King Pastoria and handed over the baby princess to Mombi. This did not please the readers, and in Ozma of Oz, although the character did not appear, the Wizard returns in the novel Dorothy and the Wizard in Oz. With Dorothy and the boy Zeb, he falls through a crack in the earth, in their journey, he acts as their guide. Oz explains that his name is Oscar Zoroaster Phadrig Isaac Norman Henkle Emmannuel Ambroise Diggs. To shorten this name, he used only his initials, but since they spell out the word pinhead, he shortened his name further, Ozma then permits him to live in Oz permanently. He becomes an apprentice to Glinda, Ozma decrees that, besides herself, only The Wizard and Glinda are allowed to use magic unless the other magic users have permits. In later books, he proves himself quite an inventor, providing devices that aid in various characters’ journeys and he introduces to Oz the use of mobile phones in Tik-Tok of Oz. Some of his most elaborate devices are the Ozpril and the Oztober, balloon-powered Ozoplanes in Ozoplaning with the Wizard of Oz, the Wizard has appeared in nearly every silent Oz film, portrayed by different actors each time. His face was also used as the projected image of the Wizard
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Oscar Diggs aka the Wizard--illustration by William Wallace Denslow (1900)
96.
Uganda
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Uganda, officially the Republic of Uganda, is a landlocked country in East Africa. It is bordered to the east by Kenya, to the north by South Sudan, to the west by the Democratic Republic of the Congo, to the south-west by Rwanda, Uganda is the worlds second most populous landlocked country after Ethiopia. The southern part of the country includes a portion of Lake Victoria, shared with Kenya. Uganda is in the African Great Lakes region, Uganda also lies within the Nile basin, and has a varied but generally a modified equatorial climate. Uganda takes its name from the Buganda kingdom, which encompasses a portion of the south of the country. The people of Uganda were hunter-gatherers until 1,700 to 2,300 years ago, beginning in 1894, the area was ruled as a protectorate by the British, who established administrative law across the territory. Uganda gained independence from Britain on 9 October 1962, luganda, a central language, is widely spoken across the country, and several other languages are also spoken including Runyoro, Runyankole, Rukiga, and Luo. The president of Uganda is Yoweri Museveni, who came to power in January 1986 after a protracted 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. According to oral tradition, the Empire of Kitara covered an important part of the lakes area, from the northern lakes Albert and Kyoga to the southern lakes Victoria. Bunyoro-Kitara is claimed as the antecedent of the Buganda, Toro, Ankole, some Luo invaded the area of Bunyoro and assimilated with the Bantu there, establishing the Babiito dynasty of the current Omukama of Bunyoro-Kitara. Arab traders moved inland from the Indian Ocean coast of East Africa in the 1830s and they were followed in the 1860s by British explorers searching for the source of the Nile. British Anglican missionaries arrived in the kingdom of Buganda in 1877 and were followed by French Catholic missionaries in 1879, the British government chartered the Imperial British East Africa Company to negotiate trade agreements in the region beginning in 1888. From 1886, there were a series of wars in Buganda. Because of civil unrest and financial burdens, IBEAC claimed that it was unable to maintain their occupation in the region, in the 1890s,32,000 labourers from British India were recruited to East Africa under indentured labour contracts to construct the Uganda Railway. Most of the surviving Indians returned home, but 6,724 decided to remain in East Africa after the lines completion, subsequently, some became traders and took control of cotton ginning and sartorial retail. British naval ships unknowingly carried rats that contained the bubonic plague and these rats spread the disease throughout Uganda. From 1900 to 1920, a sleeping sickness epidemic in the part of Uganda, along the north shores of Lake Victoria
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Left: Construction of the Owen Falls Dam in Jinja. Right: A street in Uganda in the 1950s.
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Mount Kadam, Uganda.
97.
Japan
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Japan is a sovereign island nation in Eastern Asia. Located in the Pacific Ocean, it lies off the eastern coast of the Asia Mainland and stretches from the Sea of Okhotsk in the north to the East China Sea, the kanji that make up Japans name mean sun origin. 日 can be read as ni and means sun while 本 can be read as hon, or pon, Japan is often referred to by the famous epithet Land of the Rising Sun in reference to its Japanese name. Japan is an archipelago consisting of about 6,852 islands. The four largest are Honshu, Hokkaido, Kyushu and Shikoku, the country is divided into 47 prefectures in eight regions. Hokkaido being the northernmost prefecture and Okinawa being the southernmost one, the population of 127 million is the worlds tenth largest. Japanese people make up 98. 5% of Japans total population, approximately 9.1 million people live in the city of Tokyo, the capital of Japan. Archaeological research indicates that Japan was inhabited as early as the Upper Paleolithic period, the first written mention of Japan is in Chinese history texts from the 1st century AD. Influence from other regions, mainly China, followed by periods of isolation, from the 12th century until 1868, Japan was ruled by successive feudal military shoguns who ruled in the name of the Emperor. Japan entered into a period of isolation in the early 17th century. The Second Sino-Japanese War of 1937 expanded into part of World War II in 1941, which came to an end in 1945 following the bombings of Hiroshima and Nagasaki. Japan is a member of the UN, the OECD, the G7, the G8, the country has the worlds third-largest economy by nominal GDP and the worlds fourth-largest economy by purchasing power parity. It is also the worlds fourth-largest exporter and fourth-largest importer, although Japan has officially renounced its right to declare war, it maintains a modern military with the worlds eighth-largest military budget, used for self-defense and peacekeeping roles. Japan is a country with a very high standard of living. Its population enjoys the highest life expectancy and the third lowest infant mortality rate in the world, in ancient China, Japan was called Wo 倭. It was mentioned in the third century Chinese historical text Records of the Three Kingdoms in the section for the Wei kingdom, Wa became disliked because it has the connotation of the character 矮, meaning dwarf. The 倭 kanji has been replaced with the homophone Wa, meaning harmony, the Japanese word for Japan is 日本, which is pronounced Nippon or Nihon and literally means the origin of the sun. The earliest record of the name Nihon appears in the Chinese historical records of the Tang dynasty, at the start of the seventh century, a delegation from Japan introduced their country as Nihon
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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
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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.
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Samurai could kill a commoner for the slightest insult and were widely feared by the Japanese population. Edo period, 1798
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San Marino
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Its size is just over 61 km2, with a population of 33,562. Its capital is the City of San Marino and 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 Marinus, a stonemason originating from the Roman colony on the island of Rab, in 257 CE Marinus participated in the reconstruction of Riminis city walls after their destruction by Liburnian pirates. San Marino is governed by the Constitution of San Marino, a series of six books written in Latin in the late 16th century, the country is considered to have the earliest written governing documents still in effect. The countrys economy mainly relies on finance, industry, services and 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 stable economy, with one of the lowest unemployment rates in Europe, no national debt. It is the country with more vehicles than people. Saint Marinus left the island of Arba in present-day Croatia with his lifelong friend Leo, and went to the city of Rimini as a stonemason. After the Diocletianic Persecution following his Christian sermons, he escaped to the nearby Monte Titano, the official date of the founding of what is now known as the Republic is 3 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. During the later phase of the Italian unification process in the 19th century, in recognition of this support, Giuseppe Garibaldi accepted the wish of San Marino not to be incorporated into the new Italian state. The government of San Marino made United States President Abraham Lincoln an honorary citizen and he wrote in reply, saying that the republic proved that government founded on republican principles is capable of being so administered as to be secure and enduring. Italy tried to establish a detachment of Carabinieri in the republic. Two groups of ten volunteers joined Italian forces in the fighting on the Italian front, the first as combatants, the existence of this hospital later caused Austria-Hungary to suspend diplomatic relations with San Marino. From 1923 to 1943, San Marino was under the rule of the Sammarinese Fascist Party. During World War II, San Marino remained neutral, although it was reported in an article from The New York Times that it had declared war on the United Kingdom on 17 September 1940. The Sammarinese government later transmitted a message to the British government stating that they had not declared war on the United Kingdom, Three days after the fall of Benito Mussolini in Italy, PFS rule collapsed and the new government declared neutrality in the conflict. The Fascists regained power on 1 April 1944 but kept neutrality intact, despite that, on 26 June 1944 San Marino was bombed by the Royal Air Force, in the belief that San Marino had been overrun by German forces and was being used to amass stores and ammunition
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The San Marino constitution of 1600
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The front passes Mount Titano in September 1944.
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Mount Titano
99.
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 to the north-east, Liberia to the south-east, Sierra Leone has a tropical climate, with a diverse environment ranging from savannah to rainforests. Sierra Leone has an area of 71,740 km2. Sierra Leone is divided into four regions, the Northern Province, Eastern Province, Southern Province and the Western Area. Freetown, located in the Western Area, is the capital, largest city and its economic, Bo is the second largest city, and is located in the Southern Province, about 160 miles from Freetown. Kenema, located in the Eastern Province, is the third largest city and is about 185 miles from Freetown, Koidu Town, located in the Eastern Province, is the fourth largest city, and is about 275 miles from Freetown. Makeni, located in the Northern Province, is the fifth largest of Sierra Leone five major cities, Sierra Leone is a constitutional republic with a directly elected president and a unicameral legislature. The current constitution of Sierra Leone was adopted in 1991 during the presidency of Joseph Saidu Momoh, since independent to present, Sierra Leone politics has been dominated by two major political parties, the Sierra Leone Peoples party and the All Peoples congress. The current president of SIerra Leone is Ernest Bai Koroma, a member of the APC party, the previous Sierra Leone president was Ahmad Tejan Kabbah, a member of the SLPP party, who was elected president in 1996 and won reelection for his final term in 2002. From 1991 to 2002, the Sierra Leone civil war was fought and this proxy war left more than 50,000 people dead, much of the countrys infrastructure destroyed, and over two million people displaced as refugees in neighbouring countries. In January 2002, then Sierra Leones president Ahmad Tejan Kabbah, fulfilled his promise by ending the civil war, with help by the British Government, ECOWAS. More recently, the 2014 Ebola outbreak overburdened the weak healthcare infrastructure and it created a humanitarian crisis situation and a negative spiral of weaker economic growth. The country has a low life expectancy at 57.8 years. About sixteen ethnic groups inhabit Sierra Leone, each with its own language, the two largest and most influential are the Temne and the Mende people. The Temne are predominantly found in the north of the country, the Krio language unites all the different ethnic groups in the country, especially in their trade and social interaction with each other. Sierra Leone is a predominantly Muslim country, though with an influential Christian minority, Sierra Leone is regarded as one of the most religiously tolerant nations in the world. Muslims and Christians collaborate and interact with each other peacefully, religious violence is extremely rare in the country. In politics, the majority of Sierra Leoneans vote for a candidate without regard to whether the candidate is a Muslim or a Christian
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
100.
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 French Guiana to the east, Guyana to the west, at just under 165,000 square kilometers, it is the smallest country in South America. Suriname has a population of approximately 566,000, most of live on the countrys north coast, in and around the capital and largest city. Long inhabited by cultures of indigenous tribes, Suriname was explored and contested by European powers before coming under Dutch rule in the late 17th century. In 1954, the country one of the constituent countries of the Kingdom of the Netherlands. Its indigenous peoples have been active in claiming land rights and working to preserve their traditional lands. Suriname is considered to be a culturally Caribbean country, and is a member of the Caribbean Community, while Dutch is the official language of government, business, media, and education, Sranan, an English-based creole language, is a widely used lingua franca. Suriname is the 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 ethnic, religious, and linguistic groups. This area was occupied by cultures of indigenous peoples long before European contact, remnants of which can be found in petroglyph sites at Werehpai. The name Suriname may derive from a Taino indigenous people called Surinen, British settlers, who founded the first European colony at Marshalls Creek along the Suriname River, spelled the name as Surinam. When the territory was taken over by the Dutch, it part of a group of colonies known as Dutch Guiana. The official spelling of the countrys English name was changed from Surinam to Suriname in January 1978, a notable example is Surinames national airline, Surinam Airways. The older English name is reflected in the English pronunciation, /ˈsʊrᵻnæm/ or /ˈsʊrᵻnɑːm/, in Dutch, the official language of Suriname, the pronunciation is, with the main stress on the third syllable and a schwa terminal vowel. Indigenous settlement of Suriname dates back to 3,000 BC, the largest tribes were the Arawak, a nomadic coastal tribe that lived from hunting and fishing. They were the first inhabitants in the area, the Carib also settled in the area and conquered the Arawak by using their superior sailing ships. They settled in Galibi at the mouth of the Marowijne River, while the larger Arawak and Carib tribes lived along the coast and savanna, smaller groups of indigenous peoples lived in the inland rainforest, such as the Akurio, Trió, Warrau, and Wayana. Beginning in the 16th century, French, Spanish, and English explorers visited the area, a century later, Dutch and English settlers established plantation colonies along the many rivers in the fertile Guiana plains
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
101.
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 piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are printed on special paper, show a national designation and a denomination on the front. They are sometimes a source of net profit to the issuing agency, 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, the item is then processed by the postal system, where a postmark, sometimes known as a cancellation mark, is usually applied in overlapping manner to stamp and cover. 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 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, concurrently with the first stamps, the UK offered wrappers for mail. S. Postal service for priority or express mailing, the postage stamp afforded convenience for both the mailer and postal officials, more effectively recovered costs for the postal service, and ultimately resulted in a better, faster postal system. With the conveniences stamps offered, their use resulted in greatly increased mailings during the 19th and 20th centuries, as postage stamps with their engraved imagery began to appear on a widespread basis, historians and collectors began to take notice. The study of stamps and their use is referred to as philately. Stamp collecting can be both a hobby and a form of study and reference, as government-issued postage stamps. The postage for the item was prepaid by the use of a hand-stamp to frank the mailed item. Though this stamp was applied to a letter instead of a piece of paper it is considered by many historians as the worlds first postage stamp. Rowland Hill The Englishman Sir Rowland Hill began interest in postal reform in 1835, in 1836, a Member of Parliament, Robert Wallace, provided Hill with numerous books and documents, which Hill described as a half hundred weight of material. Hill commenced a study of these documents, leading him to the 1837 publication of a pamphlet entitled Post Office Reform its Importance
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.
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Rows of perforations in a sheet of postage stamps.
102.
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 fiction, historical fiction, cyberpunk, postcyberpunk. Stephensons work explores subjects such as mathematics, cryptography, linguistics, philosophy, currency, and he also writes non-fiction articles about technology in publications such as Wired. He has also written novels with his uncle, George Jewsbury and he is currently Magic Leaps Chief Futurist. His mother worked in a laboratory, and her father was a biochemistry professor. Stephensons family moved to Champaign-Urbana, Illinois, in 1960 and then in 1966 to Ames and he graduated from Ames High School in 1977. Stephenson studied at Boston University, first specializing in physics, then switching to geography after he found that it would allow him to more time on the university mainframe. He graduated in 1981 with a B. A. in geography, since 1984, Stephenson has lived mostly in the Pacific Northwest and currently lives in Seattle with his family. Stephensons first novel, The Big U, published in 1984, was a take on life at American Megaversity. His next novel, Zodiac, was a following the exploits of a radical environmentalist protagonist in his struggle against corporate polluters. Neither novel attracted critical attention on first publication, but showcased concerns that Stephenson would further develop in his later work. Snow Crash was the first of Stephensons epic science fiction novels, in 1994, Stephenson joined with his uncle, J. Frederick George, to publish a political thriller, Interface, under the pen name Stephen Bury, they followed this in 1996 with The Cobweb. Stephensons next solo novel, published in 1995, was The Diamond Age, or A Young Ladys Illustrated Primer, seen back then as futuristic, Stephensons novel has broad range universal self-learning nanotechnology, dynabooks, extensive modern technologies, robotics, cybernetics and cyber cities. It has subsequently reissued in three separate volumes in some countries, including in French and Spanish translations. In 2013, Cryptonomicon won the Prometheus Hall of Fame Award, the Baroque Cycle is a series of historical novels set in the 17th and 18th centuries, and is in some respects a prequel to Cryptonomicon. The System of the World won the Prometheus Award in 2005, following this, Stephenson published a novel titled Anathem, a very long and detailed work, perhaps best described as speculative fiction. It is set in an Earthlike world, deals with metaphysics, Stephensons novel REAMDE was released on September 20,2011. The title is a play on the common filename README and this thriller, set in the present, centers around a group of MMORPG developers caught in the middle of Chinese cyber-criminals, Islamic terrorists, and Russian mafia
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
103.
Anathem
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Anathem is a speculative fiction novel by Neal Stephenson, published in 2008. Major themes include the interpretation of quantum mechanics and the philosophical debate between Platonic realism and nominalism. Anathem is set on and around the fictional planet Arbre, thousands of years before the events in the novel, the planets intellectuals entered concents to protect their activities from the collapse of society. The avout are allowed to communicate with people outside the walls of the concent only once every year, decade, century, or millennium, the narrator and protagonist, Fraa Erasmas, is an avout at the Concent of Saunt Edhar. His primary teacher, Fraa Orolo, discovers that a spacecraft is orbiting Arbre – a fact that the Sæcular Power attempts to cover up. Orolo secretly observes the alien ship with a camera, technology that is prohibited for the avout. Erasmas becomes aware of the content of Orolos research after Orolo is banished from the Mathic World for his possession, but the presence of the alien ship soon becomes an open secret among many of the avout at St. Edhar. Shortly after that, the Sæcular Power evoke many avout from Saunt Edhar, Erasmas and several companions, on Fraa Jads suggestion, decide to seek out Orolo. During the discussions between Orolo and Erasmas, a spacecraft lands in Orithena, on the very site of the ancient Mathic worlds Analemma. A female alien is on board, but dead of a recent gunshot wound and she has brought with her four vials of blood – presumably that of the aliens – and much evidence about their technology. Shortly thereafter, the aliens propel a massive metal rod at a nearby volcano, Orolo sacrifices his life to ensure the safety of the dead aliens remains and her blood samples, an event that leads to his canonization as Saunt Orolo. Erasmas then travels to the concent of Saunt Tredegarh – where he was expected to have gone when evoked – to attend the Convox. This is a joint conference of the avout and the Sæcular Power, dedicated to dealing with the military, political, and technical issues raised by the existence of the alien ship in Arbres orbit. Much research is done on the samples Orolo sacrificed his life to save, the mulitple-worlds interpretation of the cosmos is discussed in great detail by the high-level avout at successive evening meals to which Erasmus performs the duties of a servant. In this section of the novel, it becomes plain that Laterre is our own Earth, which serves as a higher plane of existence for Urnud and Tro. He explains that the aliens are experiencing internal conflict between two factions, the currently ruling faction intends to attack and raid Arbre for its resources in order to repair their spaceship, while the opposing faction favors open negotiation. Jules Durand offers to assist the avout of Arbre in resisting the ruling faction of the aliens, fearing alien attack after Durand has been exposed, the avout evacuate Saunt Tredegarh and all the other concents on Arbre simultaneously. Erasmas and his comrades are taken to a distant sanctuary, where they receive training for a mission to board the alien ship, three people – including Fraa Jad – are issued detonators
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
104.
Fermat's Last Theorem
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In number theory, Fermats Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n =1 and n =2 have been known to have many solutions since antiquity. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, 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. The Pythagorean equation, x2 + y2 = z2, has an number of positive integer solutions for x, y, and z. Around 1637, Fermat wrote in the margin of a book that the general equation an + bn = cn had no solutions in positive integers. Although he claimed to have a proof of his conjecture, Fermat left no details of his proof. His claim was discovered some 30 years later, after his death and this claim, which came to be known as Fermats 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, and over time Fermats Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n =4 proved, it suffices to prove the theorem for n that are prime numbers. Over the next two centuries, the conjecture was proved for only the primes 3,5, and 7, in the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and as the modularity theorem, it stood on its own and 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 Fermats Last Theorem and this potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermats Last Theorem that depended on the modularity theorem. On hearing this, English mathematician Andrew Wiles, who had a fascination with Fermats Last Theorem. In 1993, after six years working secretly on the problem, Wiless paper was massive in size and scope. A flaw was discovered in one part of his paper during peer review and required a further year and collaboration with a past student, Richard Taylor. As a result, the proof in 1995 was accompanied by a second smaller joint paper to that effect
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
105.
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, Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, in penalized regression, L1 penalty and L2 penalty refer to penalizing either the L1 norm of a solutions 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. Elastic net regularization uses a penalty term that is a combination of the L1 norm, 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, and is 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 length of a vector x = in the real vector space Rn is usually given by the Euclidean norm. The Euclidean distance between two points x and y is the length ||x − y||2 of the line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space, the class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science. For a real number p ≥1, the p-norm or Lp-norm of x is defined by ∥ x ∥ p =1 p, of course the absolute value bars are unnecessary when p is a rational number and, in reduced form, has an even numerator. The Euclidean norm from above falls into class and is the 2-norm. The L∞-norm or maximum norm is the limit of the Lp-norms for p → ∞ and it turns out that this limit is equivalent to the following definition, ∥ x ∥ ∞ = max See L-infinity. Abstractly speaking, this means that Rn together with the p-norm is a Banach space and this Banach space is the Lp-space over Rn. The grid distance or rectilinear distance between two points is never shorter than the length of the segment between them. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm, ∥ x ∥2 ≤ ∥ x ∥1. This fact generalizes to p-norms in that the p-norm ||x||p of any vector x does not grow with p, ||x||p+a ≤ ||x||p for any vector x and real numbers p ≥1
Lp space
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Unit circle (superellipse) in p = 3 / 2 norm
106.
Ptolemy's theorem
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In Euclidean geometry, Ptolemys theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy, Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. In the context of geometry, the equality is often simply written as AC·BD=AB·CD+BC·AD. Ptolemys 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. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices, Proof, Follows immediately from Ptolemys 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, if the common length of its four sides is equal to a then the length of the diagonal is equal to a 2 according to the Pythagorean theorem and the relation obviously holds. More generally, if the quadrilateral is a rectangle with sides a and b, 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 hand side of Ptolemys relation is the sum a2 + b2. A more interesting example is the relation between the length a of the side and the b of the 5 chords in a regular pentagon. In this case the relation reads b2 = a2 + ab which yields the golden ratio φ = b a =1 +52, ⇒ c = d 2 φ. whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras Theorem applied to right triangle AFD then yields b in terms of the diameter, as Copernicus wrote, The diameter of a circle being given, the sides of the triangle, tetragon, pentagon, hexagon and decagon, which the same circle circumscribes, are also given. Let ABCD be a cyclic quadrilateral, on the chord BC, the inscribed angles ∠BAC = ∠BDC, and on AB, ∠ADB = ∠ACB. Construct K on AC such that ∠ABK = ∠CBD, since ∠ABK + ∠CBK = ∠ABC = ∠CBD + ∠ABD, now, by common angles △ABK is similar to △DBC, and likewise △ABD is similar △KBC. Thus AK/AB = CD/BD, and CK/BC = DA/BD, equivalently, AK·BD = AB·CD, by adding two equalities we have AK·BD + CK·BD = AB·CD + BC·DA, and factorizing this gives ·BD = AB·CD + BC·DA. But AK+CK = AC, so AC·BD = AB·CD + BC·DA, the proof as written is only valid for simple cyclic quadrilaterals. If the quadrilateral is self-crossing then K will be located outside the line segment AC, but in this case, AK−CK=±AC, giving the expected result. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend, in what follows it is important to bear in mind that the sum of angles θ1 + θ2 + θ3 + θ4 =180 ∘. Let θ1 = θ3 and θ2 = θ4, then θ1 + θ2 = θ3 + θ4 =90 ∘
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.
107.
Rational trigonometry
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His ideas are set out in his 2005 book Divine Proportions, Rational Trigonometry to Universal Geometry. According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to some problems that occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of functions like sine and cosine by substituting their squared equivalents. To date, rational trigonometry is largely unmentioned in mainstream mathematical literature, Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary geometry. Distance is replaced with its value and 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 rational trigonometric functions, including three-dimensional volume calculations. It also deals with rational trigonometrys application to situations involving irrationals, quadrance and distance both measure separation of points in Euclidean space. Following Pythagorass theorem, the quadrance of two points A1 = and A2 = in a plane is defined as the sum of squares of differences in the x and y coordinates. In effect, the triangle inequality d1 + d2 − d3 ≥0 is modified under rational trigonometry to a symmetric form 2 ≥0 equivalent to Pythagorass theorem, spread gives one measure to the separation of two lines as a single dimensionless number in the range for Euclidean geometry. It replaces the concept of angle but has differences from angle. Trigonometric, it is the ratio for the quadrances in a right triangle. Cartesian, as a function of the three co-ordinates used to ascribe the two vectors. Linear algebra, a rational function, the square of the determinant of two vectors divided by the product of their quadrances. Suppose two lines, l1 and l2, intersect at the point A as shown at right, choose a point B ≠ A on l1 and let C be the foot of the perpendicular from B to l2. Then the spread s is s = Q Q = Q R, like angle, spread depends only on the relative slopes of two lines and is invariant under translation. Which becomes,1 − s =24 and this simplifies, in the numerator, to 2, giving,1 − s =2. Then, using the Brahmagupta–Fibonacci identity 2 +2 =, the expression for spread in terms of slopes of two lines becomes s =2. In this form a spread is the ratio of the square of a determinant of two vectors to the product of their quadrances This replaces with, with and the origin with in the previous result, s =2
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.
108.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a 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, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, 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 as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
JSTOR
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The JSTOR front page
109.
American Mathematical Society
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. 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. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L
American Mathematical Society
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American Mathematical Society
110.
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 research and information are essential to improving teaching, learning, and educational decision-making. ERIC provides access to 1.5 million bibliographic records of journal articles and other education-related materials, a key component of ERIC is its collection of grey literature in education, which is 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 using links to publisher websites and/or library holdings. Users can also access the collection through commercial vendors, statewide and institutional networks. To help users find the information they are seeking, ERIC produces a controlled vocabulary and this is a carefully selected list of education-related words and 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, various adjunct and affiliate clearinghouses, the program was consolidated into a single entity, with upgraded systems, and 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
111.
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. Recreated by the Department of Education Organization Act and signed into law by President Jimmy Carter on October 17,1979, the Department of Education Organization Act divided the Department of Health, Education, and Welfare into the Department of Education and the Department of Health and Human Services. The Department of Education is administered by the United States Secretary of Education and it has approximately 4,400 employees and an annual budget of $68 billion. The agencys official abbreviation is ED, because DOE instead refers to the United States Department of Energy and it is also often abbreviated informally as DoEd. The Department of Education does not establish schools or colleges and this has been left to state and local school districts. The Departments mission is, to student achievement and preparation for global competitiveness by fostering educational excellence. For 2006, the ED discretionary budget was $56 billion and the mandatory budget contained $23 billion, in 2009 it received additional ARRA funding of $102 billion. As of 2011, the budget is $70 billion. A previous Department of Education was created in 1867 but was demoted to an Office in 1868. As an agency not represented in the 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, in 1953, the Federal Security Agency was upgraded to cabinet-level status as the Department of Health, Education, and Welfare. In 1979, President Carter advocated for creating a cabinet-level Department of Education, carters plan was to transfer most of the Department of Health, Education, and Welfares education-related functions to the Department of Education. Carter also planned to transfer the education-related functions of the departments of Defense, Justice, Housing and Urban Development, however, many see the department as constitutional under the Commerce Clause, and that the funding role of the Department is constitutional under the Taxing and Spending Clause. The National Education Association supported the bill, while the American Federation of Teachers opposed it, as of 1979, the Office of Education had 3,000 employees and an annual budget of $12 billion. Congress appropriated to the Department of Education an annual budget of $14 billion and 17,000 employees when establishing the Department of Education, once in office, President Reagan succeeded significantly to reduce the budget. In the 1982 State of the Union Address, he pledged, after the Newt Gingrich-led revolution in 1994 had taken control of both Houses of Congress, federal control of and spending on education soared. The GOP platform read, The Federal government has no authority to be involved in school curricula or to control jobs in the market place. This is why we will abolish the Department of Education, end federal meddling in our schools, during his 1996 presidential run, Senator Bob Dole promised, Were going to cut out 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,
112.
Java (programming language)
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Java is a general-purpose computer programming language that is concurrent, class-based, object-oriented, and specifically designed to have as few implementation dependencies as possible. It is intended to let application developers write once, run anywhere, 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, 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++, the original and reference implementation Java compilers, virtual machines, and class libraries were originally released by Sun under proprietary licences. As of May 2007, in compliance with the specifications of the Java Community Process, others have also developed alternative implementations of these Sun technologies, such as the GNU Compiler for Java, GNU Classpath, and IcedTea-Web. James Gosling, Mike Sheridan, and Patrick Naughton initiated the Java language project in June 1991, Java was originally designed for interactive television, but it was too advanced for the digital cable television industry at the time. The language was initially called Oak after an oak tree that stood outside Goslings 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. Major web browsers soon incorporated the ability to run Java applets within web pages, and Java quickly became popular, while mostly outside of browsers, in January 2016, Oracle announced that Java runtime environments based on JDK9 will discontinue the browser plugin. The Java 1.0 compiler was re-written in Java by Arthur van Hoff to comply strictly with the Java 1.0 language specification, with the advent of Java 2, new versions had multiple configurations built for different types of platforms. J2EE included technologies and APIs for enterprise applications typically run in server environments, the desktop version was renamed J2SE. In 2006, for marketing purposes, Sun renamed new J2 versions as Java EE, Java ME, in 1997, Sun Microsystems approached the ISO/IEC JTC1 standards body and later the Ecma International to formalize Java, but it soon withdrew from the process. Java remains a de facto standard, controlled through the Java Community Process, at one time, Sun made most of its Java implementations available without charge, despite their proprietary software status. Sun generated revenue from Java through the selling of licenses for specialized products such as the Java Enterprise System, on November 13,2006, Sun released much of its Java virtual machine as free and open-source software, under the terms of the GNU General Public License. Suns vice-president Rich Green said that Suns ideal role with regard to Java was as an evangelist and this did not prevent Oracle from filing a lawsuit against Google shortly after that for using Java inside the Android SDK. Java software runs on everything from laptops to data centers, game consoles to scientific supercomputers, on April 2,2010, James Gosling resigned from Oracle. There were five primary goals in the creation of the Java language, It must be simple, object-oriented and it must be robust and secure
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
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Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly for documentation in libraries and increasingly also by archives, the GND is managed by the German National Library in cooperation with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license, the GND specification provides a hierarchy of high-level entities and sub-classes, useful in library classification, and an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format
Integrated Authority File
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GND screenshot
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National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. 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 in June 1948 in the present-day State Guest-House 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 merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
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