1.
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
2.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
3.
Centre (geometry)
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In geometry, a centre of an object is a point in some sense in the middle of the object. According to the definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of groups then a centre is a fixed point of all the isometries which move the object onto itself. The centre of a circle is the point equidistant from the points on the edge, similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends. For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions, so the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this being the fixed point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect. For an equilateral triangle, these are the point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex. e. F=thf for some real power h, thus the position of a centre is independent of scale. F is symmetric in its last two arguments, i. e. f= f, thus position of a centre in a triangle is the mirror-image of its position in the original triangle. This strict definition excludes pairs of points such as the Brocard points. The Encyclopedia of Triangle Centers lists over 9,000 different triangle centres, a tangential polygon has each of its sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon, a cyclic polygon has each of its vertices on a particular circle, called the circumcircle or circumscribed circle. The centre of the circumcircle, called the circumcentre, can be considered a centre of the polygon, if a polygon is both tangential and cyclic, it is called bicentric. The incentre and circumcentre of a polygon are not in general the same point. The centre of a general polygon can be defined in different ways. The vertex centroid comes from considering the polygon as being empty, the side centroid comes from considering the sides to have constant mass per unit length. The usual centre, called just the centroid comes from considering the surface of the polygon as having constant density and these three points are in general not all the same point. Centrepoint Centre of mass Chebyshev centre Fixed points of isometry groups in Euclidean space
4.
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
5.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
6.
Interior (topology)
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In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is a point of S. The interior of S is the complement of the closure of the complement of S, in this sense interior and closure are dual notions. The exterior of a set is the interior of its complement, equivalently the complement of its closure, the interior, boundary, and exterior of a subset together partition the whole space into three blocks. The interior and exterior are always open while the boundary is always closed, sets with empty interior have been called boundary sets. If S is a subset of a Euclidean space, then x is a point of S if there exists an open ball centered at x which is completely contained in S. This definition generalizes to any subset S of a metric space X with metric d, x is a point of S if there exists r >0. This definition generalises to topological spaces by replacing open ball with open set, let S be a subset of a topological space X. Then x is a point of S if x is contained in an open subset of S. The interior of a set S is the set of all points of S. The interior of S is denoted int, Int or So, the interior of a set has the following properties. Int is a subset of S. int is the union of all open sets contained in S. int is the largest open set contained in S. A set S is open if and only if S = int. int = int, if S is a subset of T, then int is a subset of int. If A is a set, then A is a subset of S if. Sometimes the second or third property above is taken as the definition of the topological interior, for more on this matter, see interior operator below. In any space, the interior of the empty set is the empty set, in any space X, if A ⊂ X, int is contained in A. If X is the Euclidean space R of real numbers, then int =, if X is the Euclidean space R, then the interior of the set Q of rational numbers is empty. If X is the complex plane C = R2, then i n t =, in any Euclidean space, the interior of any finite set is the empty set
7.
Disk (mathematics)
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In geometry, a disk is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. In Cartesian coordinates, the disk of center and radius R is given by the formula D = while the closed disk of the same center. The area of a closed or open disk of radius R is πR2, the open disk and the closed disk are not topologically equivalent, as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact, however from the viewpoint of algebraic topology they share many properties, both of them are contractible and so are homotopy equivalent to a single point. This implies that their groups are trivial, and all homology groups are trivial except the 0th one. The Euler characteristic of a point is 1, every continuous map from the closed disk to itself has at least one fixed point, this is the case n=2 of the Brouwer fixed point theorem. The statement is false for the disk, Consider for example the function f = which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the unit disk fixes every point on the half circle x 2 + y 2 =1, x >0
8.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
9.
Focus (geometry)
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In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse. An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant, a circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, a parabola is a limiting case of an ellipse in which one of the foci is a point at infinity. A hyperbola can be defined as the locus of points for each of which the value of the difference between the distances to two given foci is a constant. It is also possible to describe all conic sections in terms of a focus and a single directrix. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a positive constant. If e is zero and one the conic is an ellipse, if e=1 the conic is a parabola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero and it is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. The ellipse thus generated has its focus at the center of the directrix circle. For the parabola, the center of the moves to the point at infinity. The directrix circle becomes a curve with zero curvature, indistinguishable from a straight line. To generate a hyperbola, the radius of the circle is chosen to be less than the distance between the center of this circle and the focus, thus, the focus is outside the directrix circle. The two branches of a hyperbola are thus the two halves of a curve closed over infinity, in projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others. Plutos ellipse is entirely inside Charons ellipse, as shown in animation of the system. The barycenter is about three-quarters of the distance from Earths center to its surface, moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system
10.
Eccentricity (mathematics)
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In mathematics, the eccentricity, denoted e or ε, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular, in particular, The eccentricity of a circle is zero. The eccentricity of an ellipse which is not a circle is greater than zero, the eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1, the eccentricity of a line is infinite. Furthermore, two sections are similar if and only if they have the same eccentricity. Any conic section can be defined as the locus of points whose distances to a point and that ratio is called eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane, for β =0 the plane section is a circle, for β = α a parabola. The linear eccentricity of an ellipse or hyperbola, denoted c, is the distance between its center and either of its two foci, the eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a, that is, e = c a. The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity, the eccentricity is also sometimes called numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation, three notational conventions are in common use, e for the eccentricity and c for the linear eccentricity. ε for the eccentricity and e for the linear eccentricity, E or ϵ for the eccentricity and f for the linear eccentricity. This article uses the first notation, where, for the ellipse and the hyperbola, a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity of an ellipse is strictly less than 1, for any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis. The eccentricity is also the ratio of the axis a to the distance d from the center to the directrix. The eccentricity can be expressed in terms of the g, e = g. Define the maximum and minimum radii r max and r min as the maximum and minimum distances from either focus to the ellipse, the eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a hyperbola is 2. The eccentricity of a quadric is the eccentricity of a designated section of it
11.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
12.
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
13.
Annulus (mathematics)
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In mathematics, an annulus is a ring-shaped object, a region bounded by two concentric circles. The open annulus is topologically equivalent to both the open cylinder S1 × and the punctured plane, informally, it has the shape of a hardware washer. The area of an annulus is the difference in the areas of the circle of radius R. The area of an annulus is determined by the length of the longest line segment within the annulus, the area of an annulus sector of angle θ, with θ measured in radians, is given by A = θ2. In complex analysis an annulus ann in the plane is an open region defined as r < | z − a | < R. If r is 0, the region is known as the disk of radius R around the point a. As a subset of the plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio r/R, each annulus ann can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map z ↦ z − a R. The inner radius is then r/R <1, the Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus
14.
Concentric objects
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In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another, in the Euclidean plane, two circles that are concentric necessarily have different radii from each other. However, circles in three-dimensional space may be concentric, and have the radius as each other. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth, more generally, every two great circles on a sphere are concentric with each other and with the sphere. The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric, for the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The region of the plane between two circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell. For a given point c in the plane, the set of all circles having c as their forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii, every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of circles by a Möbius transformation. The ripples formed by dropping an object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the used in target archery or similar sports provide another familiar example of concentric circles. Coaxial cable is a type of cable in which the combined neutral. Johannes Keplers Mysterium Cosmographicum envisioned a system formed by concentric regular polyhedra. Concentric circles are found in diopter sights, a type of mechanic sights commonly found on target rifles. They usually feature a disk with a small-diametre hole near the shooters eye. When these sights are aligned, the point of impact will be in the middle of the front sight circle. Centered cube number Homoeoid Focaloid Circular symmetry Magic circle Geometry, Concentric circles demonstration With interactive animation
15.
Arc (geometry)
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In Euclidean geometry, an arc is a closed segment of a differentiable curve. A common example in the plane, is a segment of a circle called a circular arc, in space, if the arc is part of a great circle, it is called a great arc. Every pair of points on a circle determines two arcs. The length, L, of an arc of a circle with radius r and this is because L c i r c u m f e r e n c e = θ2 π. Substituting in the circumference L2 π r = θ2 π, and, with α being the angle measured in degrees, since θ = α/180π. For example, if the measure of the angle is 60 degrees and this is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The area of the sector formed by an arc and the center of a circle is A =12 r 2 θ. The area A has the proportion to the circle area as the angle θ to a full circle. We can cancel π on both sides, A r 2 = θ2, by multiplying both sides by r2, we get the final result, A =12 r 2 θ. Using the conversion described above, we find that the area of the sector for an angle measured in degrees is A = α360 π r 2. The area of the bounded by the arc and the straight line between its two end points is 12 r 2. To get the area of the arc segment, we need to subtract the area of the triangle, determined by the circles center and the two end points of the arc, from the area A. Using the intersecting chords theorem it is possible to calculate the radius r of a circle given the height H and its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W, and it is divided by the bisector into two halves, each with length W/2. The total length of the diameter is 2r, and it is divided into two parts by the first chord, the length of one part is the sagitta of the arc, H, and the other part is the remainder of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces H =2, whence 2 r − H = W24 H, so r = W28 H + H2
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Connected space
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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the topological properties that are used to distinguish topological spaces. A subset of a topological space X is a set if it is a connected space when viewed as a subspace of X. A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets, otherwise, X is said to be connected. A subset of a space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set as a space. For a topological space X the following conditions are equivalent, X is connected, X cannot be divided into two disjoint nonempty closed sets. The only subsets of X which are open and closed are X and the empty set. The only subsets of X with empty boundary are X and the empty set, X cannot be written as the union of two nonempty separated sets. All continuous functions from X to are constant, where is the two-point space endowed with the discrete topology, the maximal connected subsets of a nonempty topological space are called the connected components of the space. The components of any topological space X form a partition of X, they are disjoint, nonempty, Every component is a closed subset of the original space. It follows that, in the case where their number is finite, however, if their number is infinite, this might not be the case, for instance, the connected components of the set of the rational numbers are the one-point sets, which are not open. A space in all components are one-point sets is called totally disconnected. Clearly any totally separated space is totally disconnected, but the converse does not hold, for example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected, however, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being separated is strictly stronger than the condition of being Hausdorff. The closed interval in the subspace topology is connected, although it can, for example, be written as the union of [0, 1) and. The union of is disconnected, both of these intervals are open in the topological space
17.
Chord (geometry)
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A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just secant, is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse, a chord that passes through a circles center point is the circles diameter. Every diameter is a chord, but not every chord is a diameter, the word chord is from the Latin chorda meaning bowstring. Among properties of chords of a circle are the following, Chords are equidistant from the center if, a chord that passes through the center of a circle is called a diameter, and is the longest chord. If the line extensions of chords AB and CD intersect at a point P, the area that a circular chord cuts off is called a circular segment. The midpoints of a set of chords of an ellipse are collinear. Chords were used extensively in the development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the function for every 7.5 degrees. The circle was of diameter 120, and the lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as shown in the picture, the chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The last step uses the half-angle formula, much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them
18.
Circumference
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The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of importance in geometry and trigonometry. Informally circumference may also refer to the edge rather than to the length of the edge. The circumference of a circle is the distance around it, the term is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics and this constant, pi, is represented by the Greek letter π. The numerical value of π is 3.141592653589793, pi is defined as the ratio of a circles circumference C to its diameter d, π = C d Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference, the use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius C / r =2 π also has uses in mathematics, engineering. These uses include but are not limited to radians, computer programming, the Greek letter τ is sometimes used to represent this constant, but is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph, arc length Area Caccioppoli set Isoperimetric inequality Pythagorean theorem Volume Numericana - Circumference of an ellipse Circumference of a circle With interactive applet and animation
19.
Diameter
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In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle, both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is called the diameter. In this sense one speaks of the rather than a diameter, because all diameters of a circle or sphere have the same length. Both quantities can be calculated efficiently using rotating calipers, for a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse, the terminology is different. A diameter of an ellipse is any chord passing through the midpoint of the ellipse, for example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis, the word diameter is derived from Greek διάμετρος, diameter of a circle, from διά, across, through and μέτρον, measure. It is often abbreviated DIA, dia, d, or ⌀, the definitions given above are only valid for circles, spheres and convex shapes. However, they are cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object. The diameter of a subset of a space is the least upper bound of the set of all distances between pairs of points in the subset. So, if A is the subset, the diameter is sup, if the distance function d is viewed here as having codomain R, this implies that the diameter of the empty set equals −∞. Some authors prefer to treat the empty set as a case, assigning it a diameter equal to 0. For any solid object or set of scattered points in n-dimensional Euclidean space, in medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the supremum of the set of all distances between pairs of points in the object. In differential geometry, the diameter is an important global Riemannian invariant, the symbol or variable for diameter, ⌀, is similar in size and design to ø, the Latin small letter o with stroke. In Unicode it is defined as U+2300 ⌀ Diameter sign, on an Apple Macintosh, the diameter symbol can be entered via the character palette, where it can be found in the Technical Symbols category. The character will not display correctly, however, since many fonts do not include it. In many situations the letter ø is a substitute, which in Unicode is U+00F8 ø
20.
Circular sector
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A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the angle in radians, r the radius of the circle. A sector with the angle of 180° is called a half-disk and is bounded by a diameter. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle, the total area of a circle is πr2. Circular segment – the part of the sector which remains after removing the triangle formed by the center of the circle, conic section Gerard, L. J. V. The Elements of Geometry, in Eight Books, or, First Step in Applied Logic, London, Longmans Green, Reader & Dyer,1874
21.
Circular segment
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In geometry, a circular segment is a region of a circle which is cut off from the rest of the circle by a secant or a chord. More formally, a segment is a region of two-dimensional space that is bounded by an arc of a circle. The angle thus inscribed is θ/2 and is part of a triangle whose hypotenuse is the diameter. This is also useful in deriving some of the forms below. The area formula can be used in calculating the volume of a cylindrical tank. In the design of windows or doors with rounded tops, c and h may be the known values. Spherical segment Conic section Cross section Weisstein, Eric W. Circular segment, definition of a circular segment With interactive animation Formulae for area of a circular segment With interactive animation
22.
Secant line
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In geometry, a secant of a curve is a line that intersects two points on the curve. A chord is the interval of a secant that lies between the points at which it intersects the curve, the word secant comes from the Latin word secare, meaning to cut. A secant may be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P. As a consequence, one could say that the limit, as Q approaches P, of the secants slope, in calculus, this idea is the basis of the geometric definition of the derivative
23.
Semicircle
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In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° and it has only one line of symmetry. All lines intersecting the semicircle perpendicularly are concurrent at the center of the containing the given semicircle. A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass, if we make a semicircle with a diameter of a+b, then the length of its radius is the arithmetic mean of a and b. The length of the segment is the geometric mean, which can be proved using the Pythagorean theorem. This can be used to accomplish quadrature of a rectangle, and thus of any figure for which we can construct a rectangle of equal area, such as any polygon. The equation of a semicircle with midpoint on the diameter between its endpoints and which is concave from below is y = y 0 + r 2 −2. If it is concave from above, the equation is y = y 0 − r 2 −2. An arbelos is a region in the bounded by three semicircles connected at the corners, all on the same side of a straight line that contains their diameters. Amphitheater Archimedes twin circles Archimedes quadruplets Salinon Wigner semicircle distribution Semicircle - Mathworld
24.
Tangent
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In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
25.
Compass (drawing tool)
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A pair of compasses, also known simply as a compass, is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as tools to measure distances, Compasses can be used for mathematics, drafting, navigation and other purposes. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. Prior to computerization, compasses and other tools for manual drafting were often packaged as a bow set with interchangeable parts, today these facilities are more often provided by computer-aided design programs, so the physical tools serve mainly a didactic purpose in teaching geometry, technical drawing, etc. Compasses are usually made of metal or plastic, and consist of two connected by a hinge which can be adjusted to allow the changing of the radius of the circle drawn. Typically one part has a spike at its end, and the part a pencil. The handle is usually half a inch long. Users can grip it between their pointer finger and thumb, there are two types of legs in a pair of compasses, the straight or the steady leg and the adjustable one. Each has a purpose, the steady leg serves as the basis or support for the needle point. The screw on your hinge holds the two legs in its position, the hinge can be adjusted depending on desired stiffness, the tighter the screw the better the compass’ performance. The needle point is located on the leg, and serves as the center point of circles that are drawn. The pencil lead draws the circle on a paper or material. This holds the lead or pen in place. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, the radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass, to use a pair of compasses, place the points on a ruler and open it to the measurement of ½ of the measurement of the circle that is desired. For instance, if one desires to draw a 3 inch circle, next, place the point on the spot that you wish the center of your circle to be, and then rotate the section that has the pencil lead around the point, using the handle. Compasses-and-straightedge constructions are used to illustrate principles of plane geometry, although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles
26.
Creation myth
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A creation myth is a symbolic narrative of how the world began and how people first came to inhabit it. While in popular usage the term often refers to false or fanciful stories, formally. Cultures generally regard their creation myths as true, in the society in which it is told, a creation myth is usually regarded as conveying profound truths, metaphorically, symbolically and sometimes in a historical or literal sense. They are commonly, although not always, considered cosmogonical myths – that is, Creation myths often share a number of features. They often are considered sacred accounts and can be found in all known religious traditions. They are all stories with a plot and characters who are either deities, human-like figures, or animals and they are often set in a dim and nonspecific past that historian of religion Mircea Eliade termed in illo tempore. Creation myths develop in oral traditions and therefore typically have multiple versions, found throughout human culture, Creation myth definitions from modern references, A symbolic narrative of the beginning of the world as understood in a particular tradition and community. Creation myths are of importance for the valuation of the world, for the orientation of humans in the universe. Creation myths tell us how things began, all cultures have creation myths, they are our primary myths, the first stage in what might be called the psychic life of the species. As cultures, we identify ourselves through the collective dreams we call creation myths, … Creation myths explain in metaphorical terms our sense of who we are in the context of the world, and in so doing they reveal our real priorities, as well as our real prejudices. Our images of creation say a deal about who we are. A philosophical and theological elaboration of the myth of creation within a religious community. Religion professor Mircea Eliade defined the word myth in terms of creation, Myth narrates a history, it relates an event that took place in primordial Time. All creation myths are in one sense etiological because they attempt to explain how the world was formed, in the past historians of religion and other students of myth thought of them as forms of primitive or early-stage science or religion and analyzed them in a literal or logical sense. However they are seen as symbolic narratives which must be understood in terms of their own cultural context. Charles Long writes, The beings referred to in the myth – gods, animals, the myths should not be understood as attempts to work out a rational explanation of deity. While creation myths are not literal explications they do serve to define an orientation of humanity in the world in terms of a birth story. They are the basis of a worldview that reaffirms and guides how people relate to the world, to any assumed spiritual world
27.
Halo (religious iconography)
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A halo is a ring of light that surrounds a person in art. They have been used in the iconography of many religions to indicate holy or sacred figures, halos may be shown as almost any colour or combination of colours, but are most often depicted as golden, yellow or white when representing light or red when representing flames. Homer describes a light around the heads of heroes in battle. 450-30 BC, in the Metropolitan Museum of Art, the Colossus of Rhodes was a statue of the sun-god Helios and had his usual radiate crown. Hellenistic rulers are shown wearing radiate crowns that seem clearly to imitate this effect. The rulers of the Kushan Empire were perhaps the earliest to give themselves haloes on their coins, in Chinese and Japanese Buddhist art the halo has also been used since the earliest periods in depicting the image of Amitabha Buddha and others. Thin lines of gold often radiate outwards or inwards from the rim of the halo, elaborate haloes and especially aureoles also appear in Hindu sculpture, where they tend to develop into architectural frames in which the original idea can be hard to recognise. Theravada Buddhism and Jainism did not use the halo for many centuries, in Asian art, the nimbus is often imagined as consisting not just of light, but of flames. This type seems to first appear in Chinese bronzes of which the earliest surviving examples date from before 450 and this type is also very rarely found, and on a smaller scale, in medieval Christian art. Sometimes a thin line of flames rise up from the edges of a halo in Buddhist examples. In Tibetan paintings the flames are shown as blown by a wind. Halos are found in Islamic art from various places and periods, especially in Persian miniatures and Moghul, flaming halos derived from Buddhist art surround angels, and similar ones are often seen around Muhammad and other sacred human figures. The halo represents an aura or glow of sanctity which was conventionally drawn encircling the head, though Roman paintings have largely disappeared, save some fresco decorations, the haloed figure remains fresh in Roman mosaics. In a 2nd-century AD Roman floor mosaic preserved at Bardo, Tunisia, significantly, the triton and nereid who accompany the sea-god are not haloed. In a late 2nd century AD floor mosaic from Thysdrus, El Djem, another haloed Apollo in mosaic, from Hadrumentum, is in the museum at Sousse. The conventions of representation, head tilted, lips slightly parted, large-eyed. The halo was incorporated into Early Christian art sometime in the 4th century with the earliest iconic images of Christ, initially the only figure shown with one. At least in later Orthodox images, each bar of cross is composed of three lines, symbolising the dogmas of the Trinity, the oneness of God and the two natures of Christ
28.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
29.
Circus
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The term circus also describes the performance which has followed various formats through its 250-year modern history. Philip Astley is credited with being the father of the circus when he opened the first circus in 1768 in England. Early circuses were almost exclusively demonstrations of equestrian skills with a few types of acts to link the horsemanship performances. Performances developed significantly through the next fifty years, with large scale theatrical battle reenactments becoming a significant feature, as styles of performance have changed since the time of Astley, so too have the types of venues where these circuses have performed. The earliest modern circuses were performed in open air structures with limited covered seating, from the late 18th to late 19th century custom-made circus buildings were built with various types of seating, a centre ring, and sometimes a stage. The traditional large tents, commonly known as Big Tops were introduced in the mid 19th century as touring circuses superseded static venues and these tents eventually became the most common venue and remain so to the present day. Contemporary circuses perform in a variety of venues including tents, theatres, many circus performances are still held in a ring usually 13 m in diameter. This dimension was adopted by Philip Astley in the late 18th century as the minimum diameter that enabled a horse rider to stand upright on a cantering horse to perform their tricks. In the book De Spectaculis early Christian writer Tertullian claimed that the first circus games were staged by the goddess Circe in honour of her father Helios, the modern and commonly held idea of a circus is of a Big Top with various acts providing entertainment therein. For many, circus history begins with Englishman Philip Astley, while for others its origins go back much further—to Roman times. In Ancient Rome, the circus was a building for the exhibition of horse and chariot races, equestrian shows, staged battles, gladiatorial combat, the Roman circus buildings were, however, not circular but rectangular with semi circular ends. The lower seats were reserved for persons of rank, There were also various state boxes for the giver of the games, the circus was the only public spectacle at which men and women were not separated. It was constructed during the monarchy and, at first, built completely from wood, after being rebuilt several times, the final version of the Circus Maximus could seat 250,000 people, it was built of stone and measured 400m in length and 90m in width. Next in importance were the Circus Flaminius and the Circus Neronis, a fourth circus was constructed by Maxentius, its ruins have helped archaeologists reconstruct the Roman circus. For some time after the fall of Rome, large buildings fell out of use as centres of mass entertainment. Instead, itinerant performers, animal trainers and showmen travelled between towns throughout Europe, performing at local fairs, Astley performed stunts in a 42 ft diameter ring, which is the standard size used by circuses ever since. Astley referred to the arena as a Circle and the building as an amphitheatre. Joseph Grimaldi, the first mainstream clown, had his first major role as Little Clown in the pantomime The Triumph of Mirth, or, the Royal Circus opened in London on 4 November 1782 by Charles Dibdin and his partner Charles Hughes
30.
Arabic
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Arabic is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the lingua franca of the Arab world. Arabic is also the language of 1.7 billion Muslims. It is one of six languages of the United Nations. The modern written language is derived from the language of the Quran and it is widely taught in schools and universities, and is used to varying degrees in workplaces, government, and the media. The two formal varieties are grouped together as Literary Arabic, which is the language of 26 states. Modern Standard Arabic largely follows the standards of Quranic Arabic. Much of the new vocabulary is used to denote concepts that have arisen in the post-Quranic era, Arabic has influenced many languages around the globe throughout its history. During the Middle Ages, Literary Arabic was a vehicle of culture in Europe, especially in science, mathematics. As a result, many European languages have borrowed many words from it. Many words of Arabic origin are found in ancient languages like Latin. Balkan languages, including Greek, have acquired a significant number of Arabic words through contact with Ottoman Turkish. Arabic has also borrowed words from languages including Greek and Persian in medieval times. Arabic is a Central Semitic language, closely related to the Northwest Semitic languages, the Ancient South Arabian languages, the Semitic languages changed a great deal between Proto-Semitic and the establishment of the Central Semitic languages, particularly in grammar. Innovations of the Central Semitic languages—all maintained in Arabic—include, The conversion of the suffix-conjugated stative formation into a past tense, the conversion of the prefix-conjugated preterite-tense formation into a present tense. The elimination of other prefix-conjugated mood/aspect forms in favor of new moods formed by endings attached to the prefix-conjugation forms, the development of an internal passive. These features are evidence of descent from a hypothetical ancestor. In the southwest, various Central Semitic languages both belonging to and outside of the Ancient South Arabian family were spoken and it is also believed that the ancestors of the Modern South Arabian languages were also spoken in southern Arabia at this time. To the north, in the oases of northern Hijaz, Dadanitic and Taymanitic held some prestige as inscriptional languages, in Najd and parts of western Arabia, a language known to scholars as Thamudic C is attested
31.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
32.
Wheel
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A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the components of the wheel. Wheels, in conjunction with axles, allow heavy objects to be moved easily facilitating movement or transportation while supporting a load, wheels are also used for other purposes, such as a ships wheel, steering wheel, potters wheel and flywheel. Common examples are found in transport applications, a wheel greatly reduces friction by facilitating motion by rolling together with the use of axles. In order for wheels to rotate, a moment needs to be applied to the wheel about its axis, cognates within Indo-European include Icelandic hjól wheel, tyre, Greek κύκλος kúklos, and Sanskrit chakra, the latter both meaning circle or wheel. The invention of the falls into the late Neolithic. Note that this implies the passage of several wheel-less millennia even after the invention of agriculture and of pottery, precursors of wheels, known as tournettes or slow wheels, were known in the Middle East by the 5th millennium BCE. These were made of stone or clay and secured to the ground with a peg in the center, but required effort to turn. True potters wheels were apparently in use in Mesopotamia by 3500 BCE and possibly as early as 4000 BCE, and the oldest surviving example, which was found in Ur, dates to approximately 3100 BCE. The earliest well-dated depiction of a vehicle is on the Bronocice pot. The oldest securely dated real wheel-axle combination, that from Stare Gmajne near Ljubljana in Slovenia is now dated in 2σ-limits to 3340–3030 BCE, the axle to 3360–3045 BCE. Two types of early Neolithic European wheel and axle are known, a type of wagon construction. They both are dated to c, in China, the wheel was certainly present with the adoption of the chariot in c.1200 BCE, although Barbieri-Low argues for earlier Chinese wheeled vehicles, c.2000 BC. In Britain, a wooden wheel, measuring about 1 m in diameter, was uncovered at the Must Farm site in East Anglia in 2016. The specimen, dating from 1, 100–800 years BCE, represents the most complete, the wheels hub is also present. A horses spine found nearby suggests the wheel may have part of a horse-drawn cart. The wheel was found in a settlement built on stilts over wetland and it is thought that the primary obstacle to large-scale development of the wheel in the Americas was the absence of domesticated large animals which could be used to pull wheeled carriages. The only large animal that was domesticated in the Western hemisphere, Nubians from after about 400 BCE used wheels for spinning pottery and as water wheels
33.
Gear
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A gear or cogwheel is a rotating machine part having cut teeth, or cogs, which mesh with another toothed part to transmit torque. Geared devices can change the speed, torque, and direction of a power source, Gears almost always produce a change in torque, creating a mechanical advantage, through their gear ratio, and thus may be considered a simple machine. The teeth on the two meshing gears all have the same shape, two or more meshing gears, working in a sequence, are called a gear train or a transmission. A gear can mesh with a linear toothed part, called a rack, the gears in a transmission are analogous to the wheels in a crossed, belt pulley system. An advantage of gears is that the teeth of a gear prevent slippage, in transmissions with multiple gear ratios—such as bicycles, motorcycles, and cars—the term gear as in first gear refers to a gear ratio rather than an actual physical gear. The term describes similar devices, even when the ratio is continuous rather than discrete, or when the device does not actually contain gears. Early examples of gears date from the 4th century BC in China, examples of further development include, Ma Jun used gears as part of a south-pointing chariot. The Antikythera mechanism is an example of an early and intricate geared device. Its time of construction is now estimated between 150 and 100 BC, the water-powered grain-mill, the water-powered saw mill, fulling mill, and other applications of watermill often used gears. The first mechanical clocks were built in AD725, the 1386 Salisbury cathedral clock may be the worlds oldest working mechanical clock. The definite ratio that teeth give gears provides an advantage over other drives in precision machines such as watches that depend upon an exact velocity ratio, an external gear is one with the teeth formed on the outer surface of a cylinder or cone. Conversely, a gear is one with the teeth formed on the inner surface of a cylinder or cone. For bevel gears, a gear is one with the pitch angle exceeding 90 degrees. Internal gears do not cause output shaft direction reversal, spur gears or straight-cut gears are the simplest type of gear. They consist of a cylinder or disk with teeth projecting radially, though the teeth are not straight-sided, the edge of each tooth is straight and aligned parallel to the axis of rotation. These gears mesh together correctly only if fitted to parallel shafts, No axial thrust is created by the tooth loads. Spur gears are excellent at moderate speeds but tend to be noisy at high speeds, helical or dry fixed gears offer a refinement over spur gears. The leading edges of the teeth are not parallel to the axis of rotation, since the gear is curved, this angling makes the tooth shape a segment of a helix
34.
Science
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Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations, disciplines which use science, like engineering and medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, chemistry, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge. In particular, it was the type of knowledge which people can communicate to each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle later created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science
35.
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
36.
Astrology and astronomy
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Astrology and astronomy were archaically treated together, and were only gradually separated in Western 17th century philosophy with the rejection of astrology. During the later part of the period, astronomy was treated as the foundation upon which astrology could operate. Since the 18th century they have come to be regarded as separate disciplines. Astronomy, the study of objects and phenomena originating beyond the Earths atmosphere, is a science and is a widely studied academic discipline, in pre-modern times, most cultures did not make a clear distinction between the two disciplines, putting them both together as one. This overlap does not mean that astrology and astronomy were always regarded as one, in ancient Greece, pre-Socratic thinkers such as Anaximander, Xenophanes, Anaximenes, and Heraclides speculated about the nature and substance of the stars and planets. Astronomers such as Eudoxus observed planetary motions and cycles, and created a geocentric cosmological model that would be accepted by Aristotle and this model generally lasted until Ptolemy, who added epicycles to explain the retrograde motion of Mars. The Platonic school promoted the study of astronomy as a part of philosophy because the motions of the heavens demonstrate an orderly, in the third century BC, Babylonian astrology began to make its presence felt in Greece. Astrology was criticized by Hellenistic philosophers such as the Academic Skeptic Carneades, however, the notions of the Great Year and eternal recurrence were Stoic doctrines that made divination and fatalism possible. In the Hellenistic world, the Greek words astrologia and astronomia were often used interchangeably, plato taught about astronomia and stipulated that planetary phenomena should be described by a geometrical model. The first solution was proposed by Eudoxus, Aristotle favored a physical approach and adopted the word astrologia. Eccentrics and epicycles came to be thought of as useful fictions, for a more general public, the distinguishing principle was not evident and either word was acceptable. For the Babylonian horoscopic practice, the words used were apotelesma and katarche. Isidore identified the two strands entangled in the discipline and called them astrologia naturalis and astrologia superstitiosa. Astrology was widely accepted in medieval Europe as astrological texts from Hellenistic and Arabic astrologers were translated into Latin, in the late Middle Ages, its acceptance or rejection often depended on its reception in the royal courts of Europe. Not until the time of Francis Bacon was astrology rejected as a part of scholastic metaphysics rather than empirical observation, because of their lengthy shared history, it sometimes happens that the two are confused with one another even today. Many contemporary astrologers, however, do not claim that astrology is a science, but think of it as a form of divination like the I-Ching, the primary goal of astronomy is to understand the physics of the universe. Astrologers use astronomical calculations for the positions of bodies along the ecliptic and attempt to correlate celestial events with earthly events. Astronomers consistently use the method, naturalistic presuppositions and abstract mathematical reasoning to investigate or explain phenomena in the universe
37.
History of science in the Middle Ages
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The history of science is the study of the development of science and scientific knowledge, including both the natural sciences and social sciences. Historiography of science, in contrast, studies the methods by which historians study the history of science, the English word scientist is relatively recent—first coined by William Whewell in the 19th century. Previously, people investigating nature called themselves natural philosophers, traditionally, historians of science have defined science sufficiently broadly to include those earlier inquiries. In prehistoric times, advice and knowledge was passed from generation to generation in an oral tradition, for example, the domestication of maize for agriculture has been dated to about 9,000 years ago in southern Mexico, before the development of writing systems. Similarly, archaeological evidence indicates the development of knowledge in preliterate societies. The development of writing enabled knowledge to be stored and communicated across generations with much greater fidelity, many ancient civilizations collected astronomical information in a systematic manner through simple observation. Though they had no knowledge of the physical structure of the planets and stars. Basic facts about human physiology were known in places. Considerable observation of flora and fauna was also performed. Ancient Egypt made significant advances in astronomy, mathematics and medicine and their development of geometry was a necessary outgrowth of surveying to preserve the layout and ownership of farmland, which was flooded annually by the Nile river. The 3-4-5 right triangle and other rules of geometry were used to build structures. Egypt was also a center of research for much of the Mediterranean. However, while Egyptian medicine had some practices, it was not without its ineffective. Medical historians believe that ancient Egyptian pharmacology, for example, was largely ineffective, E. R. Lloyd played a significant role in the development of this methodology. The Ebers papyrus also contains evidence of traditional empiricism, from their beginnings in Sumer around 3500 BC, the Mesopotamian people began to attempt to record some observations of the world with numerical data. But their observations and measurements were taken for purposes other than for elucidating scientific laws. A concrete instance of Pythagoras law was recorded, as early as the 18th century BC, dated 1900 BC, possibly millennia before Pythagoras, but an abstract formulation of the Pythagorean theorem was not. In Babylonian astronomy, records of the motions of the stars, planets, even today, astronomical periods identified by Mesopotamian proto-scientists are still widely used in Western calendars such as the solar year and the lunar month
38.
Rhind Mathematical Papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian and it dates to around 1550 BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus, the Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and it was copied by the scribe Ahmes, from a now-lost text from the reign of king Amenemhat III. Written in the script, this Egyptian manuscript is 33 cm tall. The papyrus began to be transliterated and mathematically translated in the late 19th century, the mathematical translation aspect remains incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later historical note on its verso likely dating from the period of his successor, Khamudi. In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving Accurate reckoning for inquiring into things, the scribe Ahmose writes this copy. Several books and articles about the Rhind Mathematical Papyrus have been published, a more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. The first part of the Rhind papyrus consists of reference tables, the problems start out with simple fractional expressions, followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table, the fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. For example,2 /15 =1 /10 +1 /30. The decomposition of 2/n into unit fractions is never more than 4 terms long as in for example 2 /101 =1 /101 +1 /202 +1 /303 +1 /606. This table is followed by a smaller, tiny table of fractional expressions for the numbers 1 through 9 divided by 10. Problems 1-7, 7B and 8-40 are concerned with arithmetic and elementary algebra, problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 = 7/4 and 1 + 2/3 + 1/3 =2 by different fractions, problems 21–23 are problems in completion, which in modern notation are simply subtraction problems. Problems 24–34 are ‘’aha’’ problems, these are linear equations, problem 32 for instance corresponds to solving x + 1/3 x + 1/4 x =2 for x. Problems 35–38 involve divisions of the heqat, which is an ancient Egyptian unit of volume, problems 39 and 40 compute the division of loaves and use arithmetic progressions
39.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
40.
Tughrul Tower
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Tuğrul Tower is a 12th-century monument, located in the city of Rey, Iran. Tuğrul Tower is near Rashkan Castle, the 20 meters tall brick tower is the tomb of Seljuk ruler Tuğrul Beg, who died in Rey in 1063. Originally, like other monuments of its time, it was capped by a conical dome, the dome collapsed during an earthquake. The thickness of the walls varies from 1.75 to 2.75 meters, the inner and outer diameters are 11 and 16 meters, respectively. The exterior shape is that of a polygon with 24 angles in its design, at the top of the tower Kufic inscriptions were originally observable. Naser al-Din Shah ordered some restorations to be made to the top part of the tower, the tower is protected by Irans Cultural Heritage Organization. Architecture of Iran Toqrol Tower, Ray
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