Conch

Conch is a common name applied to a number of different medium to large-sized shells. The term applies to large snails whose shell has a high spire and a noticeable siphonal canal. In North America, a conch is identified as a queen conch, indigenous to the waters of the Bahamas. Queen conchs are valued for seafood, are used as fish bait; the group of conchs that are sometimes referred to as "true conchs" are marine gastropod molluscs in the family Strombidae in the genus Strombus and other related genera. For example, see Lobatus gigas, the queen conch, Laevistrombus canarium, the dog conch. Many other species are often called "conch", but are not at all related to the family Strombidae, including Melongena species, the horse conch Triplofusus papillosus. Species referred to as conchs include the sacred chank or more shankha shell and other Turbinella species in the family Turbinellidae; the English word "conch" is attested in Middle English, coming from Latin concha, which in turn comes from Greek konchē from Proto-Indo-European root *konkho-, cognate with Sanskrit śaṅkha.

The meat of conchs is eaten raw in salads, or cooked, as in burgers, chowders and gumbos. All parts of the conch meat are edible. Conch is most indigenous to the Bahamas, is served in fritter and soup forms. Conch is eaten in the West Indies. Restaurants all over the islands serve this particular meat. In the Dominican Republic and Haiti, conch is eaten in curries or in a spicy soup, it is locally referred to as lambi. In the Turks and Caicos Islands, the Annual Conch Festival is held in November each year, located at the Three Queens Bar/Restaurant in Blue Hills. Local restaurateurs compete for the best and most original conch dishes, which are judged by international chefs. Free sampling of the dishes follows the judging. In Puerto Rico, conch is served as a ceviche called ensalada de carrucho, consisting of raw conch marinated in lime juice, olive oil, garlic, green peppers, onions, it is used to fill empanadas. In Panama, conch is known as cambombia and is served as ceviche de cambombia consisting of raw conch marinated in lime juice, chopped onions, finely chopped habaneros, vinegar.

Conch is popular in Italy and among Italian Americans. Called scungille, it is eaten in a variety of ways, but most in salads or cooked in a sauce for pasta, it is included as one of the dishes prepared for the Feast of the Seven Fishes. In East Asian cuisines, this seafood is cut into thin slices and steamed or stir-fried. Eighty-percent of the queen conch meat in international trade is imported into the United States; the Florida Keys were a major source of queen conchs until the 1970s, but the conchs are now scarce and all harvesting of them in Florida waters is prohibited. Conch shells can be used as wind instruments, they are prepared by cutting a hole in the spire of the shell near the apex, blowing into the shell as if it were a trumpet, as in blowing horn. Sometimes, a mouthpiece is used. Pitch is adjusted by moving one's hand out of the aperture. Various species of large marine gastropod shells can be turned into "blowing shells", but some of the best-known species used are the sacred chank or shankha Turbinella pyrum, the Triton's trumpet Charonia tritonis, the queen conch Strombus gigas.

Many different kinds of mollusks can produce pearls. Pearls from the queen conch, S. gigas, are rare and have been collectors' items since Victorian times. Conch pearls occur in a range of hues, including white and orange, with many intermediate shades, but pink is the colour most associated with the conch pearl, such that these pearls are sometimes referred to as "pink pearls". In some gemological texts, non-nacreous gastropod pearls used to be referred to as "calcareous concretions" because they were "porcellaneous", rather than "nacreous", sometimes known as "orient"; the GIA and CIBJO now use the term "pearl"—or, where appropriate, the more descriptive term "non-nacreous pearl"—when referring to such items, under Federal Trade Commission rules, various mollusc pearls may be referred to as "pearls" without qualification. Although not nacreous, the surfaces of fine conch pearls have a unique and attractive appearance of their own; the microstructure of conch pearls comprises aligned bundles of microcrystalline fibres that create a shimmering iridescent effect known as "flame structure".

The effect is a form of chatoyancy, caused by the interaction of light rays with the microcrystals in the pearl's surface, it somewhat resembles moiré silk. Conch shells are sometimes used as decoration, as decorative planters, in cameo making. In classic Maya art, conchs are shown being used in many ways, including as paint and ink holders for elite scribes, as bugles or trumpets, as hand weapons. Conch shells have been used as shell money in several cultures; some American Aboriginals used cylindrical conch columella beads as part of breastplates and other personal adornment. In India, the Bengali bride-to-be is adorned with conch shell and coral bangles

Curve

In mathematics, a curve is speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line. 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. A curve is a topological space, 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, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is its ending point—that is, a path from any of its points to the same point. Related meanings include the graph of a function and a two-dimensional graph. Interest in curves began; 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, for example by a stick in the sand on a beach. 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 today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length", while a straight line is defined as "a line that lies evenly with the points on itself". Euclid's idea of a line is clarified by the statement "The extremities of a line are points,". Commentators further classified lines according to various schemes. For example: Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction; these curves include: The conic sections studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles and used as a method to double the cube.

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 and square the circle; 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; this enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, those that cannot, transcendental curves. Curves had been described as "geometrical" or "mechanical" according to how they were, or could be, generated. Conic sections were applied in astronomy by Kepler. Newton worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways.

The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into'ovals'; the statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, differential geometry; the era of the space-filling curves provoked the modern definitions of curve. In general, a curve is defined through a continuous function γ: I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ, called a curve. In general topology, when non-differentiable functions are considered, it is the map γ, called a curve, because its image may look differently from what is called a curve.

For example, the image of the Peano curve fills the square. On the other hand, when one considers curves defined by a differentiable function, this is the image of the function, called a curve; the curve is said to be simple, or a Jordan arc, if γ is injective, i.e. if for all x, y in I, we have γ = γ

Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object. For example, the equations x = cos t y = sin t form a parametric representation of the unit circle, where t is the parameter: A point is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: =. Parametric representations are nonunique, so the same quantities may be expressed by a number of different parameterizations. In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, the number of equations being equal to the dimension of the space in which the manifold or variety is considered.

Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is labeled t. Parameterizations are non-unique. In kinematics, objects' paths through space are described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter. Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position; such parametric curves can be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as r = its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of parametric equations is in the field of computer-aided design. For example, consider the following three representations, all of which are used to describe planar curves; the first two types are known as non-parametric, representations of curves.

In particular, the non-parametric representation depends on the choice of the coordinate system and does not lend itself well to geometric transformations, such as rotations and scaling. These problems can be addressed by rewriting the non-parametric equations in parametric form. Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides a, b and their hypotenuse c are coprime integers; as a and b are not both one may exchange them to have a and the parameterization is a = 2 m n, b = m 2 − n 2, c = m 2 + n 2, where the parameters m and n are positive coprime integers that are not both odd. By multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of parametric equations to a single implicit equation involves eliminating the variable t from the simultaneous equations x = f, y = g

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. 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, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study 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 means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

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, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Nicomedes (mathematician)

Nicomedes was an ancient Greek mathematician. Nothing is known about Nicomedes' life apart from references in his works. Studies have stated that Nicomedes was born in about 280 BC and died in about 210 BC, it is known that he lived around the time of Eratosthenes or after, because he criticized Eratosthenes' method of doubling the cube. It is known that Apollonius of Perga called a curve of his creation a "sister of the conchoid", suggesting that he was naming it after Nicomedes' famous curve, it is believed that Nicomedes lived after Eratosthenes and before Apollonius of Perga. Like many geometers of the time, Nicomedes was engaged in trying to solve the problems of doubling the cube and trisecting the angle, both problems we now understand to be impossible using the tools of classical geometry. In the course of his investigations, Nicomedes created the conchoid of Nicomedes. Nicomedes discovered three distinct types of conchoids, now unknown. Pappus wrote: "Nicomedes trisected any rectilinear angle by means of the conchoidal curves, the construction and properties of which he handed down, being himself the discoverer of their peculiar character".

Nicomedes used the Hippias' quadratrix to square the circle, since according to Pappus, "For the squaring of the circle there was used by Dinostratus and certain other persons a certain curve which took its name from this property, for it is called by them square-forming". Eutocius mentions that Nicomedes "prided himself inordinately on his discovery of this curve, contrasting it with Eratosthenes's mechanism for finding any number of mean proportionals, to which he objected formally and at length on the ground that it was impracticable and outside the spirit of geometry". T. L. Heath, A History of Greek Mathematics. G. J. Toomer, Biography in Dictionary of Scientific Biography. O'Connor, John J..

Encyclopædia Britannica, Eleventh Edition

The Encyclopædia Britannica, Eleventh Edition is a 29-volume reference work, an edition of the Encyclopædia Britannica. It was developed during the encyclopaedia's transition from a British to an American publication; some of its articles were written by the best-known scholars of the time. This edition of the encyclopedia, containing 40,000 entries, is now in the public domain, many of its articles have been used as a basis for articles in Wikipedia. However, the outdated nature of some of its content makes its use as a source for modern scholarship problematic; some articles have special value and interest to modern scholars as cultural artifacts of the 19th and early 20th centuries. The 1911 eleventh edition was assembled with the management of American publisher Horace Everett Hooper. Hugh Chisholm, who had edited the previous edition, was appointed editor in chief, with Walter Alison Phillips as his principal assistant editor. Hooper bought the rights to the 25-volume 9th edition and persuaded the British newspaper The Times to issue its reprint, with eleven additional volumes as the tenth edition, published in 1902.

Hooper's association with The Times ceased in 1909, he negotiated with the Cambridge University Press to publish the 29-volume eleventh edition. Though it is perceived as a quintessentially British work, the eleventh edition had substantial American influences, not only in the increased amount of American and Canadian content, but in the efforts made to make it more popular. American marketing methods assisted sales; some 14% of the contributors were from North America, a New York office was established to coordinate their work. The initials of the encyclopedia's contributors appear at the end of selected articles or at the end of a section in the case of longer articles, such as that on China, a key is given in each volume to these initials; some articles were written by the best-known scholars of the time, such as Edmund Gosse, J. B. Bury, Algernon Charles Swinburne, John Muir, Peter Kropotkin, T. H. Huxley, James Hopwood Jeans and William Michael Rossetti. Among the lesser-known contributors were some who would become distinguished, such as Ernest Rutherford and Bertrand Russell.

Many articles were carried over from some with minimal updating. Some of the book-length articles were divided into smaller parts for easier reference, yet others much abridged; the best-known authors contributed only a single article or part of an article. Most of the work was done by British Museum scholars and other scholars; the 1911 edition was the first edition of the encyclopædia to include more than just a handful of female contributors, with 34 women contributing articles to the edition. The eleventh edition introduced a number of changes of the format of the Britannica, it was the first to be published complete, instead of the previous method of volumes being released as they were ready. The print type was subject to continual updating until publication, it was the first edition of Britannica to be issued with a comprehensive index volume in, added a categorical index, where like topics were listed. It was the first not to include long treatise-length articles. Though the overall length of the work was about the same as that of its predecessor, the number of articles had increased from 17,000 to 40,000.

It was the first edition of Britannica to include biographies of living people. Sixteen maps of the famous 9th edition of Stielers Handatlas were translated to English, converted to Imperial units, printed in Gotha, Germany by Justus Perthes and became part this edition. Editions only included Perthes' great maps as low quality reproductions. According to Coleman and Simmons, the content of the encyclopedia was distributed as follows: Hooper sold the rights to Sears Roebuck of Chicago in 1920, completing the Britannica's transition to becoming a American publication. In 1922, an additional three volumes, were published, covering the events of the intervening years, including World War I. These, together with a reprint of the eleventh edition, formed the twelfth edition of the work. A similar thirteenth edition, consisting of three volumes plus a reprint of the twelfth edition, was published in 1926, so the twelfth and thirteenth editions were related to the eleventh edition and shared much of the same content.

However, it became apparent that a more thorough update of the work was required. The fourteenth edition, published in 1929, was revised, with much text eliminated or abridged to make room for new topics; the eleventh edition was the basis of every version of the Encyclopædia Britannica until the new fifteenth edition was published in 1974, using modern information presentation. The eleventh edition's articles are still of value and interest to modern readers and scholars as a cultural artifact: the British Empire was at its maximum, imperialism was unchallenged, much of the world was still ruled by monarchs, the tragedy of the modern world wars was still in the future, they are an invaluable resource for topics omitted from modern encyclopedias for biography and the history of science and technology. As a literary text, the encyclopedia has value as an example of early 20th-century prose. For example, it employs literary devices, such as pathetic fallacy, which are not as common in modern reference texts.

In 1917, using the pseudonym of S. S. Van Dine, the US art critic and author Willard Huntington Wright published Misinforming a Nation, a 200+

Conchoid of Dürer

The conchoid of Dürer called Dürer's shell curve, is a variant of a conchoid or plane algebraic curve, named after Albrecht Dürer. It is not a true conchoid. Suppose two perpendicular lines are given, with intersection point O. For concreteness we may assume that O is the origin, that is. Let points Q = and R = move on the axes in such a way that q + r = b, a constant. On the line QR, extended as necessary, mark points P and P' at a fixed distance a from Q; the locus of the points P and P' is Dürer's conchoid. The equation of the conchoid in Cartesian form is 2 y 2 − 2 b y 2 + y 2 − a 2 x 2 + 2 a 2 b + a 2 = 0. In parametric form the equation is given by x = b cos cos − sin + a cos , y = a sin , where the parameter t is measured in radians; the curve has two components, asymptotic to the lines y = ± a / 2. Each component is a rational curve. If a > b there is a loop. Special cases include: a = 0: the line y = 0, it was first described by the German painter and mathematician Albrecht Dürer in his book Underweysung der Messung, calling it Ein muschellini.

Dürer only drew one branch of the curve. Conchoid of de Sluze List of curves Weisstein, Eric W. "Dürer's Conchoid". MathWorld