1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
2.
Shape
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A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons, examples of geons include cones and spheres. Some simple shapes can be put into broad categories, for instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into categories, triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces, ellipsoids, which are egg-shaped or sphere-shaped objects, cylinders, and cones. If an object falls into one of these categories exactly or even approximately, thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. Similarity, Two objects are similar if one can be transformed into the other by a scaling, together with a sequence of rotations, translations. Isotopy, Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the b and d are a reflection of each other, and hence they are congruent and similar. Sometimes, only the outline or external boundary of the object is considered to determine its shape, for instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic objects such as a point, a line, a curve, a plane. However, most shapes occurring in the world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean space have the shape if one can be transformed to the other by a combination of translations, rotations. In other words, the shape of a set of points is all the information that is invariant to translations, rotations
3.
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
4.
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
5.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
6.
Hyperrectangle
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In geometry, an n-orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals. A three-dimensional orthotope is also called a rectangular prism, rectangular cuboid. A special case of an n-orthotope, where all edges are equal length, is the n-cube, the dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex, rhombic n-fusil, or n-lozenge. It is constructed by 2n points located in the center of the rectangular faces. An n-fusils Schläfli symbol can be represented by a sum of n line segments. A 1-fusil is a line segment and its plane cross selections in all pairs of axes are rhombi. Minimum bounding box Coxeter, Harold Scott MacDonald
7.
Aspect ratio (image)
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The aspect ratio of an image describes the proportional relationship between its width and its height. It is commonly expressed as two separated by a colon, as in 16,9.5 yards high. The most common ratios used today in the presentation of films in cinemas are 1.85,1 and 2.39,1. Two common videographic aspect ratios are 4,3, the video format of the 20th century. Other cinema and video aspect ratios exist, but are used infrequently, in still camera photography, the most common aspect ratios are 4,3,3,2, and more recently being found in consumer cameras 16,9. Other aspect ratios, such as 5,3,5,4, in motion picture formats, the physical size of the film area between the sprocket perforations determines the images size. The universal standard is a frame that is four perforations high, the film itself is 35 mm wide, but the area between the perforations is 24.89 mm×18.67 mm, leaving the de facto ratio of 4,3, or 1.33,1. A4,3 ratio mimics human eyesight visual angle of 155°h x 120°v, the motion picture industry convention assigns a value of 1.0 to the image’s height, an anamorphic frame is often incorrectly described as 2.40,1 or 2.40. After 1952, a number of aspect ratios were experimented with for anamorphic productions, a SMPTE specification for anamorphic projection from 1957 finally standardized the aperture to 2.35,1. An update in 1970 changed the ratio to 2.39,1 in order to make splices less noticeable. This aspect ratio of 2.39,1 was confirmed by the most recent revision from August 1993, in American cinemas, the common projection ratios are 1.85,1 and 2.39,1. Some European countries have 1.66,1 as the wide screen standard, the Academy ratio of 1.375,1 was used for all cinema films in the sound era until 1953. During that time, television, which had a aspect ratio of 1.33,1. Hollywood responded by creating a number of wide-screen formats, CinemaScope, Todd-AO. The flat 1.85,1 aspect ratio was introduced in May 1953, and became one of the most common cinema projection standards in the U. S. and elsewhere. The goal of these lenses and aspect ratios was to capture as much of the frame as possible, onto as large an area of the film as possible. In either case the image was squeezed horizontally to fit the frame size. Development of various film camera systems must ultimately cater to the placement of the frame in relation to the constraints of the perforations
8.
Display aspect ratio
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The aspect ratio of a computer display is the proportional relationship between its width and its height. It is expressed as two separated by a colon. Current common aspect ratios for displays are 5,4,4,3,16,10 and 16,9, computer displays with aspect ratio wider than 4,3 are also called widescreen. Widescreen computer displays are typically of the 16,9 or 16,10 aspect ratio, in 2008, the computer industry started to move over from 4,3 and 16,10 to 16,9. Until about 2003, most computer monitors had a 4,3 aspect ratio, between 2003 and 2006, monitors with 16,10 aspect ratio became commonly available, first in laptops and later also in standalone computer monitors. Reasons for this transition was productive uses for such monitors, i. e.16,10 became the most common aspect ratio for widescreen computer monitors until 2008. In 2008, the industry started to move from 4,3 and 16,10 to 16,9 as the standard aspect ratio for monitors. By 2010, virtually all computer monitor and laptop manufacturers had moved to the 16,9 aspect ratio. In 2011, non-widescreen displays with 4,3 aspect ratios still were being manufactured and he also predicted that by the end of 2011, production on all 4,3 or similar panels will be halted due to a lack of demand. In March 2011, the 16,9 resolution of 1920×1080 became the most commonly used resolution among Steam users, in April 2012, the 16,9 resolution of 1366×768 became the most commonly used resolution worldwide, the previous most common resolution was 1024×768. The third most common resolution at the time was 1280×800, since 2011, several monitors complying with the Digital Cinema Initiatives 4K standard have been produced. The standard specifies a resolution of 4096×2160 and a ratio of almost 1.9,1. Since 2005 most video games are made for the 16,9 aspect ratio and 16,9 computer displays therefore offer the best compatibility. 16,9 video games are letterboxed on a 16,10 or 4,3 display or have reduced field of view,4,3 monitors have the best compatibility with older games released prior to 2005 when that aspect ratio was the mainstream standard for computer displays. Movies usually are in 2.39,1,16,9 or 1.85,1,16,9 computer displays have the best compatibility. During the 2000s, DVDs and TV broadcasts shifted from 4,3 to 16,9, today the TV and DVDs are in 16,9 or 1.85,1 format,16,9 displays are optimal for their playback on a computer. 16,9 material on a 16,10 or 4,3 display will be letterboxed, in data processing or viewing 4,3 material such as older films, older TV broadcasts or older digital photographs, the widescreen will be letterboxed. Microsoft recommends a 16,9 display for computers running Windows 8
9.
Paper size
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Many paper size standards conventions have existed at different times and in different countries. Today, the A and B series of ISO216, which includes the commonly used A4 size, are the standard used by almost every country. In some areas under strong US influence, Letter is more prevalent, Paper sizes affect writing paper, stationery, cards, and some printed documents. The international standard for envelopes is the C series of ISO269, the international paper size standard is ISO216. It is based on the German DIN476 standard for paper sizes, ISO paper sizes are all based on a single aspect ratio of square root of 2, or approximately 1,1.4142. There are different series, as well as several extensions, the following international paper sizes are included in Cascading Style Sheets, A3, A4, A5, B4, B5. The base A0 size of paper is defined as having an area of 1 m2, rounded to the nearest millimetre, that is 841 by 1,189 millimetres. Successive paper sizes in the series A1, A2, A3 and this also effectively halves the area of each sheet. The most frequently used paper size is A4 measuring 210 by 297 millimetres, folded brochures of any size can be made by using sheets of the next larger size, e. g. A4 sheets are folded to make A5 brochures. The system allows scaling without compromising the aspect ratio from one size to another—as provided by office photocopiers, similarly, two sheets of A4 can be scaled down and fit exactly on 1 sheet without any cutoff or margins. The behavior of the ratio is easily proven, on a sheet of paper, let a be the long side and b be the short side, thus. When the sheet of paper is folded in half widthwise, let c be the length of the new short side, c = a/2. If we take the ratio of the folded paper we have, b c = b a 2 =2 a b =22 =2 Therefore. The advantages of basing a paper size upon an aspect ratio of √2 were first noted in 1786 by the German scientist and philosopher Georg Christoph Lichtenberg. The formats that became A2, A3, B3, B4 and B5 were developed in France on proposition of the mathematician Lazare Carnot, early in the 20th century, Dr Walter Porstmann turned Lichtenbergs idea into a proper system of different paper sizes. Porstmanns system was introduced as a DIN standard in Germany in 1922, even today, the paper sizes are called DIN A4 in everyday use in Germany and Austria. The DIN476 standard spread quickly to other countries, by 1977, A4 was the standard letter format in 88 of 148 countries. Today the standard has been adopted by all countries in the world except the United States, in Mexico, Costa Rica, Colombia, Venezuela, Chile, and the Philippines, the US letter format is still in common use, despite their official adoption of the ISO standard
