1.
Quadrilateral
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In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle
2.
Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. 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 edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
3.
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
4.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
5.
Convex polygon
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A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a polygon whose interior is a convex set. In a convex polygon, all angles are less than or equal to 180 degrees. A simple polygon which is not convex is called concave, the following properties of a simple polygon are all equivalent to convexity, Every internal angle is less than or equal to 180 degrees. Every point on line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is contained in a closed half-plane defined by each of its edges. For each edge, the points are all on the same side of the line that the edge defines. The angle at each vertex contains all vertices in its edges. The polygon is the hull of its edges. Additional properties of convex polygons include, The intersection of two convex polygons is a convex polygon, a convex polygon may br triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Hellys theorem, For every collection of at least three convex polygons, if the intersection of three of them is nonempty, then the whole collection has a nonempty intersection. Krein–Milman theorem, A convex polygon is the hull of its vertices. Thus it is defined by the set of its vertices. Hyperplane separation theorem, Any two convex polygons with no points in common have a separator line, if the polygons are closed and at least one of them is compact, then there are even two parallel separator lines. Inscribed triangle property, Of all triangles contained in a convex polygon, inscribing triangle property, every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.5 × Area ≤ Area ≤2 × Area, the mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the perimeter as the polygon. Every polygon inscribed in a circle, if not self-intersecting, is convex, however, not every convex polygon can be inscribed in a circle
6.
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
7.
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
8.
Canadian English
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Canadian English is the set of varieties of the English language native to Canada. A larger number,28 million people, reported using English as their dominant language, 82% of Canadians outside the province of Quebec reported speaking English natively, but within Quebec the figure was just 7. 7% as most of its residents are native speakers of Quebec French. Canadian English contains elements of British English and American English, as well as many Canadianisms, the construction of identities and English-language varieties across political borders is a complex social phenomenon. The term Canadian English is first attested in a speech by the Reverend A. Constable Geikie in an address to the Canadian Institute in 1857, Canadian English is the product of five waves of immigration and settlement over a period of more than two centuries. Studies on earlier forms of English in Canada are rare, yet connections with other work to historical linguistics can be forged, an overview of diachronic work on Canadian English, or diachronically-relevant work, is Dollinger. Until the 2000s, basically all commentators on the history of CanE have argued from the language-external history, an exception has been in the area of lexis, where Avis et als Dictionary of Canadianisms on Historical Principles, offered real-time historical data though its quotations. Recently, historical linguists have started to study earlier Canadian English on historical linguistic data, dCHP-1 is now available in open access. )Most notably, Dollinger pioneered the historical corpus linguistic approach for English in Canada with CONTE and offers a developmental scenario for 18th and 19th century Ontario. Recently, Reuter, with a 19th-century newspaper corpus from Ontario, has confirmed the scenario laid out in Dollinger, Canadian spelling of the English language combines British and American conventions. Words such as realize and paralyze are usually spelled with -ize or -yze rather than -ise or -yse, french-derived words that in American English end with -or and -er, such as color or center, often retain British spellings. While the United States uses the Anglo-French spelling defense and offense, some nouns, as in British English, take -ice while matching verbs take -ise – for example, practice and licence are nouns while practise and license are the respective corresponding verbs. Canadian spelling sometimes retains the British practice of doubling consonants when adding suffixes to words even when the syllable is not stressed. Compare Canadian travelled, counselling, and marvellous to American traveled, counseling, in American English, such consonants are only doubled when stressed, thus, for instance, controllable and enthralling are universal. In other cases, Canadians and Americans differ from British spelling, such as in the case of nouns like curb and tire, Canadian spelling conventions can be partly explained by Canadas trade history. For instance, the British spelling of the word cheque probably relates to Canadas once-important ties to British financial institutions, Canadas political history has also had an influence on Canadian spelling. Canadas first prime minister, John A. Macdonald, once directed the Governor General of Canada to issue an order-in-council directing that government papers be written in the British style, a contemporary reference for formal Canadian spelling is the spelling used for Hansard transcripts of the Parliament of Canada. Many Canadian editors, though, use the Canadian Oxford Dictionary, often along with the chapter on spelling in Editing Canadian English, and, throughout part of the 20th century, some Canadian newspapers adopted American spellings, for example, color as opposed to the British-based colour. Some of the most substantial historical spelling data can be found in Dollinger, the use of such spellings was the long-standing practice of the Canadian Press perhaps since that news agencys inception, but visibly the norm prior to World War II. The practice of dropping the letter u in such words was also considered a labour-saving technique during the days of printing in which movable type was set manually
9.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
10.
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
11.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
12.
Trapezoidal rule
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In mathematics, and more specifically in numerical analysis, the trapezoidal rule is a technique for approximating the definite integral ∫ a b f d x. The trapezoidal rule works by approximating the region under the graph of the function f as a trapezoid and it follows that ∫ a b f d x ≈. A2016 paper reports that the rule was in use in Babylon before 50 BC for integrating the velocity of Jupiter along the ecliptic. The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, however for various classes of rougher functions, the trapezoidal rule has faster convergence in general than Simpsons rule. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, for a domain discretized into N equally spaced panels, or N+1 grid points a = x1 < x2 <. < xN+1 = b, where the spacing is h = N the approximation to the integral becomes ∫ a b f d x ≈ h 2 ∑ k =1 N = b − a 2 N. When the grid spacing is non-uniform, one can use the formula ∫ a b f d x ≈12 ∑ k =1 N and this can also be seen from the geometric picture, the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, if the interval of the integral being approximated includes an inflection point, the error is harder to identify. Further terms in this error estimate are given by the Euler–Maclaurin summation formula and it is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions. The trapezoidal rule often converges very quickly for periodic functions, in the error formula above, f = f, and only the O term remains. More detailed analysis can be found in, for various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpsons rule
13.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
14.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
15.
Rectangle
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In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
16.
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
17.
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
18.
Isosceles trapezoid
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In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a case of a trapezoid. In any isosceles trapezoid two opposite sides are parallel, and the two sides are of equal length. The diagonals are also of equal length, the base angles of an isosceles trapezoid are equal in measure. Rectangles and squares are usually considered to be cases of isosceles trapezoids though some sources would exclude them. Another special case is a 3-equal side trapezoid, sometimes known as a trapezoid or a trisosceles trapezoid. They can also be dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices. Any non-self-crossing quadrilateral with one axis of symmetry must be either an isosceles trapezoid or a kite. Every antiparallelogram has a trapezoid as its convex hull, and may be formed from the diagonals. The base angles have the same measure, the segment that joins the midpoints of the parallel sides is perpendicular to them. Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals, the diagonals divide each other into segments with lengths that are pairwise equal, in terms of the picture below, AE = DE, BE = CE. In an isosceles trapezoid the base angles have the same measure pairwise, in the picture below, angles ∠ABC and ∠DCB are obtuse angles of the same measure, while angles ∠BAD and ∠CDA are acute angles, also of the same measure. Since the lines AD and BC are parallel, angles adjacent to opposite bases are supplementary, the diagonals of an isosceles trapezoid have the same length, that is, every isosceles trapezoid is an equidiagonal quadrilateral. Moreover, the diagonals divide each other in the same proportions, as pictured, the diagonals AC and BD have the same length and divide each other into segments of the same length. The ratio in each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect. The height is, according to the Pythagorean theorem, given by h = p 2 −2 =124 c 2 −2. The distance from point E to base AD is given by d = a h a + b where a and b are the lengths of the parallel sides AD and BC, and h is the height of the trapezoid. The area of a trapezoid is equal to the average of the lengths of the base
19.
Reflection symmetry
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Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry, in 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its image is called mirror symmetric. The set of operations that preserve a property of the object form a group. Two objects are symmetric to each other with respect to a group of operations if one is obtained from the other by some of the operations. Another way to think about the function is that if the shape were to be folded in half over the axis. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match, a circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles, quadrilaterals with reflection symmetry are kites, deltoids, rhombuses, and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges, for an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between 2/3 and 1 for any convex shape, for each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically C2. The fundamental domain is a half-plane or half-space, in certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry, in contexts in modern physics the term parity or P-symmetry is used for both. For more general types of reflection there are more general types of reflection symmetry. For example, with respect to a non-isometric affine involution with respect to circle inversion, most animals are bilaterally symmetric, likely because this supports forward movement and streamlining. Mirror symmetry is used in architecture, as in the facade of Santa Maria Novella. It is also found in the design of ancient structures such as Stonehenge, Symmetry was a core element in some styles of architecture, such as Palladianism. Patterns in nature Point reflection symmetry Stewart, Ian, weidenfeld & Nicolson. is potty Weyl, Hermann. Mapping with symmetry - source in Delphi Reflection Symmetry Examples from Math Is Fun
20.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
21.
Point reflection
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Not to be confused with inversive geometry, in which inversion is through a circle instead of a point. In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space, point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has one fixed point. It is equivalent to a transformation with scale factor equal to -1. The point of inversion is called homothetic center. The term reflection is loose, and considered by some an abuse of language, with preferred, however. Such maps are involutions, meaning that they have order 2 – they are their own inverse, in dimension 1 these coincide, as a point is a hyperplane in the line. In terms of algebra, assuming the origin is fixed. Reflection in a hyperplane has a single −1 eigenvalue, while point reflection has only the −1 eigenvalue. The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across a plane perpendicular to the axis of rotation, in dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is R e f p =2 p − a, in the case where p is the origin, point reflection is simply the negation of the vector a. In Euclidean geometry, the inversion of a point X with respect to a point P is a point X* such that P is the midpoint of the segment with endpoints X. In other words, the vector from X to P is the same as the vector from P to X*, the formula for the inversion in P is x*=2a−x where a, x and x* are the position vectors of P, X and X* respectively. This mapping is an isometric involutive affine transformation which has one fixed point. When the inversion point P coincides with the origin, point reflection is equivalent to a case of uniform scaling. This is an example of linear transformation, when P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation, homothety with homothetic center coinciding with P, and scale factor = -1. This is an example of non-linear affine transformation), the composition of two point reflections is a translation
22.
Tangential trapezoid
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It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the sides are called the bases. The legs can be equal, but they dont have to be, examples of tangential trapezoids are rhombi and squares. A convex quadrilateral is tangential if and only if opposite sides satisfy Pitots theorem, in turn, a tangential quadrilateral is a trapezoid if and only if either of the following two properties hold, It has two adjacent angles that are supplementary. Specifically, a tangential quadrilateral ABCD is a trapezoid with parallel bases AB and CD if, the product of two adjacent tangent lengths equals the product of the other two tangent lengths. The formula for the area of a trapezoid can be simplified using Pitots theorem to get a formula for the area of a tangential trapezoid. If the bases have lengths a and b, and any one of the two sides has length c, then the area K is given by the formula K = a + b | b − a | a b. The area can be expressed in terms of the tangent lengths e, f, g, h as K = e f g h 4. Using the same notations as for the area, the radius in the incircle is r = K a + b = a b | b − a |, the diameter of the incircle is equal to the height of the tangential trapezoid. The inradius can also be expressed in terms of the tangent lengths as r = e f g h 4. Moreover, if the tangent lengths e, f, g, h emanate respectively from vertices A, B, C, D and AB is parallel to DC, then r = e h = f g. If the incircle is tangent to the bases at P and Q, then P, I and Q are collinear, the angles AID and BIC in a tangential trapezoid ABCD, with bases AB and DC, are right angles. The incenter lies on the median, the median of a tangential trapezoid equals one fourth of the perimeter of the trapezoid. It also equals half the sum of the bases, as in all trapezoids, if two circles are drawn, each with a diameter coinciding with the legs of a tangential trapezoid, then these two circles are tangent to each other. A right tangential trapezoid is a trapezoid where two adjacent angles are right angles. If the bases have lengths a and b, then the inradius is r = a b a + b, thus the diameter of the incircle is the harmonic mean of the bases. The right tangential trapezoid has the area K = a b, an isosceles tangential trapezoid is a tangential trapezoid where the legs are equal. Since an isosceles trapezoid is cyclic, a tangential trapezoid is a bicentric quadrilateral
23.
Tangential quadrilateral
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In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter, Tangential quadrilaterals are a special case of tangential polygons. Due to the risk of confusion with a quadrilateral that has a circumcircle, all triangles have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle, the section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have an incircle. Examples of tangential quadrilaterals are the kites, which include the rhombi, the kites are exactly the tangential quadrilaterals that are also orthodiagonal. A right kite is a kite with a circumcircle, if a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral, and if it is both tangential and a trapezoid, it is called a tangential trapezoid. In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle, conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter. Conversely a convex quadrilateral in which a + c = b + d must be tangential, the second of these is almost the same as one of the equalities in Urquharts theorem. The only differences are the signs on both sides, in Urquharts theorem there are instead of differences. Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if, a characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. Several more characterizations are known in the four subtriangles formed by the diagonals, the eight tangent lengths of a tangential quadrilateral are the line segments from a vertex to the points where the incircle is tangent to the sides. From each vertex there are two congruent tangent lengths, the two tangency chords of a tangential quadrilateral are the line segments that connect points on opposite sides where the incircle is tangent to these sides. These are also the diagonals of the contact quadrilateral, the area K of a tangential quadrilateral is given by K = r ⋅ s, where s is the semiperimeter and r is the inradius. Another formula is K =12 p 2 q 2 −2 which gives the area in terms of the p, q. The area can also be expressed in terms of just the four tangent lengths, if these are e, f, g, h, then the tangential quadrilateral has the area K =. Furthermore, the area of a quadrilateral can be expressed in terms of the sides a, b, c, d. Since eg = fh if and only if the quadrilateral is also cyclic and hence bicentric. For given side lengths, the area is maximum when the quadrilateral is also cyclic, then K = a b c d since opposite angles are supplementary angles
24.
Saccheri quadrilateral
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A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, for a Saccheri quadrilateral ABCD, the sides AD and BC are equal in length and perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles, as it turns out, when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclids fifth postulate. When the summit angles are acute, this leads to hyperbolic geometry, and when the summit angles are obtuse. Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory and he did show that the obtuse case was contradictory, but failed to properly handle the acute case. Saccheri quadrilaterals were first considered by Omar Khayyam in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid. Let ABCD be a Saccheri quadrilateral having AB as base, CD as summit and CA, the following properties are valid in any Saccheri quadrilateral in hyperbolic geometry, The summit angles are equal and acute. The summit is longer than the base, two Saccheri quadrilaterals are congruent if, the base segments and summit angles are congruent the summit segments and summit angles are congruent. The line segment joining the midpoints of the sides is not perpendicular to either side, besides the 2 right angles, these quadrilaterals have acute summit angles. The tilings exhibit a symmetry, and include, Lambert quadrilateral Coxeter. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag,1975
25.
Lambert quadrilateral
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In geometry, a Lambert quadrilateral, named after Johann Heinrich Lambert, is a quadrilateral in which three of its angles are right angles. It is now known that the type of the angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle, a Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so half of the Saccheri quadrilateral is a Lambert quadrilateral
26.
Ex-tangential quadrilateral
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In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius, the excenter lies at the intersection of six angle bisectors. The ex-tangential quadrilateral is closely related to the tangential quadrilateral, another name for an excircle is an escribed circle, but that name has also been used for a circle tangent to one side of a convex quadrilateral and the extensions of the adjacent two sides. In that context all convex quadrilaterals have four escribed circles, kites are examples of ex-tangential quadrilaterals. Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths, a convex quadrilateral is ex-tangential if and only if there are six concurrent angles bisectors. This is possible in two different ways—either as a + b = c + d or a + d = b + c and this was proved by Jakob Steiner in 1846. These equations are related to the Pitot theorem for tangential quadrilaterals. If opposite sides in a convex quadrilateral ABCD intersect at E and F, the implication to the right is named after L. M. Urquhart although it was proved long before by Augustus De Morgan in 1841. Daniel Pedoe named it the most elementary theorem in Euclidean geometry since it only concerns straight lines and distances and that there in fact is an equivalence was proved by Mowaffac Hajja, which makes the equality to the right another necessary and sufficient condition for a quadrilateral to be ex-tangential. A few of the metric characterizations of tangential quadrilaterals have very similar counterparts for ex-tangential quadrilaterals, thus a convex quadrilateral has an incircle or an excircle outside the appropriate vertex if and only if any one of the five necessary and sufficient conditions below is satisfied. The notations in this table are as follows, In a convex quadrilateral ABCD, an ex-tangential quadrilateral ABCD with sides a, b, c, d has area K = a b c d sin B + D2. Note that this is the formula as the one for the area of a tangential quadrilateral. The exradius for a quadrilateral with consecutive sides a, b, c, d is given by r = K | a − c | = K | b − d | where K is the area of the quadrilateral. For an ex-tangential quadrilateral with sides, the exradius is maximum when the quadrilateral is also cyclic. These formulas explain why all parallelograms have infinite exradius, if an ex-tangential quadrilateral also has a circumcircle, it is called an ex-bicentric quadrilateral. Then, since it has two opposite angles, its area is given by K = a b c d which is the same as for a bicentric quadrilateral. If x is the distance between the circumcenter and the excenter, then 12 +12 =1 r 2 and this is the same equation as Fusss theorem for a bicentric quadrilateral
27.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
28.
Degree (angle)
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A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
29.
Diagonal
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In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal, in matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, non-mathematical uses, diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or on a diagonal, hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the cross over the poles at an angle. In association football, the system of control is the method referees. As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices, therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, in a convex polygon, if no three diagonals are concurrent at a single point, the number of regions that the diagonals divide the interior into is given by + =24. The number of regions is 1,4,11,25,50,91,154,246, in a polygon with n angles the number of diagonals is given by n ∗2. The number of intersections between the diagonals is given by, in the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, the off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero, a superdiagonal entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those A i j with j = i and this plays an important part in geometry, for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly and this is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1,1,0,0,0, a geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion to. Topics In Algebra, Waltham, Blaisdell Publishing Company, ISBN 978-1114541016 Nering, linear Algebra and Matrix Theory, New York, Wiley, LCCN76091646 Diagonals of a polygon with interactive animation Polygon diagonal from MathWorld. Diagonal of a matrix from MathWorld
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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
31.
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
32.
Collinearity
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In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear, in greater generality, the term has been used for aligned objects, that is, things being in a line or in a row. In any geometry, the set of points on a line are said to be collinear, in Euclidean geometry this relation is intuitively visualized by points lying in a row on a straight line. However, in most geometries a line is typically an object type. For instance, in geometry, where lines are represented in the standard model by great circles of a sphere. Such points do not lie on a line in the Euclidean sense. A mapping of a geometry to itself which sends lines to lines is called a collineation, the linear maps of vector spaces, viewed as geometric maps, map lines to lines, that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation, the de Longchamps point also has other collinearities. Any vertex, the tangency of the side with an excircle. Any vertex, the tangency of the side with the incircle. From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle, the lines connecting the feet of the altitudes intersect the opposite sides at collinear points. A triangles incenter, the midpoint of an altitude, and the point of contact of the side with the excircle relative to that side are collinear. The incenter, the centroid, and the Spieker circles center are collinear, the circumcenter, the Brocard midpoint, and the Lemoine point of a triangle are collinear. Two perpendicular lines intersecting at the orthocenter of a triangle intersect each of the triangles extended sides. The midpoints on the three sides of these points of intersection are collinear in the Droz–Farny line. In a convex quadrilateral ABCD whose opposite sides intersect at E and F, the midpoints of AC, BD, and EF are collinear, if the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. In a convex quadrilateral, the quasiorthocenter H, the area centroid G, and the quasicircumcenter O are collinear in this order, other collinearities of a tangential quadrilateral are given in Tangential quadrilateral#Collinear points. In a cyclic quadrilateral, the circumcenter, the centroid
33.
Midpoint
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In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the both of the segment and of the endpoints. The midpoint of a segment in n-dimensional space whose endpoints are A = and B = is given by A + B2 and that is, the ith coordinate of the midpoint is a i + b i 2. Given two points of interest, finding the midpoint of the segment they determine can be accomplished by a compass. The point where the line connecting the cusps intersects the segment is then the midpoint of the segment and it is more challenging to locate the midpoint using only a compass, but it is still possible according to the Mohr-Mascheroni theorem. The midpoint of any diameter of a circle is the center of the circle, any line perpendicular to any chord of a circle and passing through its midpoint also passes through the circles center. The midpoint of any segment which is an area bisector or perimeter bisector of an ellipse is the ellipses center, the ellipses center is also the midpoint of a segment connecting the two foci of the ellipse. The midpoint of a segment connecting a hyperbolas vertices is the center of the hyperbola, the perpendicular bisector of a side of a triangle is the line that is perpendicular to that side and passes through its midpoint. The three perpendicular bisectors of a triangles three sides intersect at the circumcenter, the median of a triangles side passes through both the sides midpoint and the triangles opposite vertex. The three medians of a triangle intersect at the triangles centroid, the nine-point center of a triangle lies at the midpoint between the circumcenter and the orthocenter. These points are all on the Euler line, the medial triangle of a given triangle has vertices at the midpoints of the given triangles sides. It shares the same centroid and medians with the given triangle, the perimeter of the medial triangle equals the semiperimeter of the original triangle, and its area is one quarter of the area of the original triangle. The orthocenter of the medial triangle coincides with the circumcenter of the original triangle, every triangle has an inscribed ellipse, called its Steiner inellipse, that is internally tangent to the triangle at the midpoints of all its sides. This ellipse is centered at the centroid, and it has the largest area of any ellipse inscribed in the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse, the two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the segment joining the midpoints of the diagonals are concurrent at a point called the vertex centroid. The four maltitudes of a quadrilateral are the perpendiculars to a side through the midpoint of the opposite side. If the quadrilateral is cyclic, these maltitudes all meet at a point called the anticenter
34.
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
35.
Arithmetic mean
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In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, the term arithmetic mean is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the mean is used frequently in fields such as economics, sociology, and history. For example, per capita income is the average income of a nations population. While the arithmetic mean is used to report central tendencies, it is not a robust statistic. In a more obscure usage, any sequence of values that form a sequence between two numbers x and y can be called arithmetic means between x and y. The arithmetic mean is the most commonly used and readily understood measure of central tendency, in statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the values of each. For example, let us consider the monthly salary of 10 employees of a firm,2500,2700,2400,2300,2550,2650,2750,2450,2600,2400. The arithmetic mean is 2500 +2700 +2400 +2300 +2550 +2650 +2750 +2450 +2600 +240010 =2530, If the data set is a statistical population, then the mean of that population is called the population mean. If the data set is a sample, we call the statistic resulting from this calculation a sample mean. The arithmetic mean of a variable is denoted by a bar, for example as in x ¯. The arithmetic mean has several properties that make it useful, especially as a measure of central tendency and these include, If numbers x 1, …, x n have mean x ¯, then + ⋯ + =0. The mean is the single number for which the residuals sum to zero. If the arithmetic mean of a population of numbers is desired, the arithmetic mean may be contrasted with the median. The median is defined such that half the values are larger than, and half are smaller than, If elements in the sample data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample 1,2,3,4, the average is 2.5, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as 1,2,4,8,16, in this case, the arithmetic average is 6.2 and the median is 4
36.
Aryabhata
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Aryabhata or Aryabhata I was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya and the Arya-siddhanta, furthermore, in most instances Aryabhatta would not fit the metre either. Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga and this corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra, Bhāskara I describes Aryabhata as āśmakīya, one belonging to the Aśmaka country. During the Buddhas time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India. It has been claimed that the aśmaka where Aryabhata originated may be the present day Kodungallur which was the capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr, however, K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence. Aryabhata mentions Lanka on several occasions in the Aryabhatiya, but his Lanka is an abstraction and it is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I, identify Kusumapura as Pāṭaliputra, Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. Aryabhata is the author of treatises on mathematics and astronomy. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and it also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, a third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, direct details of Aryabhatas work are known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra and it is also occasionally referred to as Arya-shatas-aShTa, because there are 108 verses in the text. It is written in the terse style typical of sutra literature
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Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
38.
Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
39.
Indian mathematics
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Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama, the decimal number system in worldwide use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, in addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China and this was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solution. In the prose section, the form was not considered so important as the ideas involved, all mathematical works were orally transmitted until approximately 500 BCE, thereafter, they were transmitted both orally and in manuscript form. A later landmark in Indian mathematics was the development of the series expansions for functions by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series. However, they did not formulate a theory of differentiation and integration. Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of practical mathematics. The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4,2,1, considered favourable for the stability of a brick structure. They used a system of weights based on the ratios, 1/20, 1/10, 1/5, 1/2,1,2,5,10,20,50,100,200. They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, the inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length was divided into ten equal parts, bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The religious texts of the Vedic Period provide evidence for the use of large numbers, by the time of the Yajurvedasaṃhitā-, numbers as high as 1012 were being included in the texts. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta, With three-fourths Puruṣa went up, the Satapatha Brahmana contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras list rules for the construction of fire altars. Most mathematical problems considered in the Śulba Sūtras spring from a single theological requirement, according to, the Śulba Sūtras contain the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately and they contain lists of Pythagorean triples, which are particular cases of Diophantine equations
40.
Indian astronomy
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Indian astronomy has a long history stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley Civilization or earlier, astronomy later developed as a discipline of Vedanga or one of the auxiliary disciplines associated with the study of the Vedas, dating 1500 BCE or older. The oldest known text is the Vedanga Jyotisha, dated to 1400–1200 BCE, as with other traditions, the original application of astronomy was thus religious. Indian astronomy flowered in the 5th-6th century, with Aryabhata, whose Aryabhatiya represented the pinnacle of astronomical knowledge at the time, Later the Indian astronomy significantly influenced Muslim astronomy, Chinese astronomy, European astronomy, and others. Other astronomers of the era who further elaborated on Aryabhatas work include Brahmagupta, Varahamihira. Some of the earliest forms of astronomy can be dated to the period of Indus Valley Civilization or earlier, some cosmological concepts are present in the Vedas, as are notions of the movement of heavenly bodies and the course of the year. Thus, the Shulba Sutras, texts dedicated to altar construction, discusses advanced mathematics, Vedanga Jyotisha is another of the earliest known Indian texts on astronomy, it includes the details about the sun, moon, nakshatras, lunisolar calendar. Greek astronomical ideas began to enter India in the 4th century BCE following the conquests of Alexander the Great, by the early centuries of the Common Era, Indo-Greek influence on the astronomical tradition is visible, with texts such as the Yavanajataka and Romaka Siddhanta. Later astronomers mention the existence of various siddhantas during this period and these were not fixed texts but rather an oral tradition of knowledge, and their content is not extant. The text today known as Surya Siddhanta dates to the Gupta period and was received by Aryabhata, the classical era of Indian astronomy begins in the late Gupta era, in the 5th to 6th centuries. The Pañcasiddhāntikā by Varāhamihira approximates the method for determination of the direction from any three positions of the shadow using a gnomon. By the time of Aryabhata the motion of planets was treated to be rather than circular. The divisions of the year were on the basis of religious rites, in the Vedānga Jyotiṣa, the year begins with the winter solstice. Hindu calendars have several eras, The Hindu calendar, counting from the start of the Kali Yuga, has its epoch on 18 February 3102 BCE Julian, the Vikrama Samvat calendar, introduced about the 12th century, counts from 56–57 BCE. The Saka Era, used in some Hindu calendars and in the Indian national calendar, has its epoch near the equinox of year 78. The Saptarshi calendar traditionally has its epoch at 3076 BCE and this device finds mention in the works of Varāhamihira, Āryabhata, Bhāskara, Brahmagupta, among others. The Cross-staff, known as Yasti-yantra, was used by the time of Bhaskara II and this device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. The clepsydra was used in India for astronomical purposes until recent times, Ōhashi notes that, Several astronomers also described water-driven instruments such as the model of fighting sheep
41.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
42.
Heron's formula
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Herons formula states that the area of a triangle whose sides have lengths a, b, and c is A = s, where s is the semiperimeter of the triangle, that is, s = a + b + c 2. Herons formula can also be written as A =14 A =142 − A =142 −2 A =144 a 2 b 2 −2. Let △ABC be the triangle sides a =4, b =13. The semiperimeter is s = 1/2 = 1/2 =16, in this example, the side lengths and area are all integers, making it a Heronian triangle. However, Herons formula works well in cases where one or all of these numbers is not an integer. The formula is credited to Heron of Alexandria, and a proof can be found in his book, Metrica, written c. A formula equivalent to Herons, namely A =12 a 2 c 2 −2 and it was published in Shushu Jiuzhang, written by Qin Jiushao and published in 1247. Herons original proof made use of quadrilaterals, while other arguments appeal to trigonometry as below, or to the incenter. A modern proof, which uses algebra and is quite unlike the one provided by Heron, let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. The difference of two squares factorization was used in two different steps, the following proof is very similar to one given by Raifaizen. By the Pythagorean theorem we have b2 = h2 + d2, subtracting these yields a2 − b2 = c2 − 2cd. Herons formula as given above is numerically unstable for triangles with a small angle when using floating point arithmetic. A stable alternative involves arranging the lengths of the sides so that a ≥ b ≥ c, the brackets in the above formula are required in order to prevent numerical instability in the evaluation. Three other area formulas have the structure as Herons formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, Herons formula is a special case of Brahmaguptas formula for the area of a cyclic quadrilateral. Herons formula and Brahmaguptas formula are both special cases of Bretschneiders formula for the area of a quadrilateral, Herons formula can be obtained from Brahmaguptas formula or Bretschneiders formula by setting one of the sides of the quadrilateral to zero. Herons formula is also a case of the formula for the area of a trapezoid or trapezium based only on its sides. Herons formula is obtained by setting the smaller parallel side to zero, another generalization of Herons formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins
43.
Semiperimeter
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In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles. When the semiperimeter occurs as part of a formula, it is denoted by the letter s. The semiperimeter is used most often for triangles, the formula for the semiperimeter of a triangle with side lengths a, b, the three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two each of whose length equals the semiperimeter. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle, a line through the triangles incenter bisects the perimeter if and only if it also bisects the area. A triangles semiperimeter equals the perimeter of its medial triangle, by the triangle inequality, the longest side length of a triangle is less than the semiperimeter. The area A of any triangle is the product of its inradius and its semiperimeter, the area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Herons formula, A = s. The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths and this formula can be derived from the law of sines. The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is t a =2 b c s b + c, in a right triangle, the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius, the area of the right triangle is where a and b are the legs. The formula for the semiperimeter of a quadrilateral with side lengths a, b, c and d is s = a + b + c + d 2. The simplest form of Brahmaguptas formula for the area of a quadrilateral has a form similar to that of Herons formula for the triangle area. Bretschneiders formula generalizes this to all convex quadrilaterals, K = − a b c d ⋅ cos 2 , the four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius. The area of a regular polygon is the product of its semiperimeter
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Brahmagupta's formula
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In Euclidean geometry, Brahmaguptas formula finds the area of any cyclic quadrilateral given the lengths of the sides. Brahmaguptas formula gives the area K of a quadrilateral whose sides have lengths a, b, c, d as K = where s. This formula generalizes Herons formula for the area of a triangle, a triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle, if the semiperimeter is not used, Brahmaguptas formula is K =14. Another equivalent version is K =2 +8 a b c d −24 ⋅ Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of △ADB and △BDC, but since ABCD is a cyclic quadrilateral, ∠DAB = 180° − ∠DCB. Therefore, K =12 p q sin A +12 r s sin A K2 =142 sin 2 A4 K2 =2 =2 −2 cos 2 A. Solving for common side DB, in △ADB and △BDC, the law of cosines gives p 2 + q 2 −2 p q cos A = r 2 + s 2 −2 r s cos C. Substituting cos C = −cos A and rearranging, we have 2 cos A = p 2 + q 2 − r 2 − s 2. Substituting this in the equation for the area,4 K2 =2 −14216 K2 =42 −2. The right-hand side is of the form a2 − b2 = and hence can be written as which, upon rearranging the terms in the square brackets, introducing the semiperimeter S = p + q + r + s/2,16 K2 =16. Taking the square root, we get K =, an alternative, non-trigonometric proof utilizes two applications of Herons triangle area formula on similar triangles. This more general formula is known as Bretschneiders formula and it is a property of cyclic quadrilaterals that opposite angles of a quadrilateral sum to 180°. It follows from the equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths. A related formula, which was proved by Coolidge, also gives the area of a convex quadrilateral. It is K = −14 where p and q are the lengths of the diagonals of the quadrilateral, in a cyclic quadrilateral, pq = ac + bd according to Ptolemys theorem, and the formula of Coolidge reduces to Brahmaguptas formula. Herons formula for the area of a triangle is the case obtained by taking d =0. The relationship between the general and extended form of Brahmaguptas formula is similar to how the law of cosines extends the Pythagorean theorem, increasingly complicated closed-form formulas exist for the area of general polygons on circles, as described by Maley et al