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Lune (mathematics)
In plane geometry, a lune () is the concave-convex region bounded by two circular arcs. It has one boundary portion for which the connecting segment of any two nearby points moves outside the region and another boundary portion for which the connecting segment of any two nearby points lies entirely inside the region. A convex-convex region is termed a lens. Formally, a lune is the relative complement of one disk in another (where they intersect but neither is a subset of the other). Alternatively, if A and B are disks, then A \smallsetminus A \cap B is a lune. Squaring the lune In the 5th century BC, Hippocrates of Chios showed that the Lune of Hippocrates and two other lunes could be exactly squared (converted into a square having the same area) by straightedge and compass. In 1766 the Finnish mathematician Daniel Wijnquist, quoting Daniel Bernoulli, listed all five geometrical squareable lunes, adding to those known by Hippocrates. In 1771 Leonard Euler gave a general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luno Ebena Geom 1
Luno may refer to: * "Luno" (song), a song on Bloc Party's 2005 album ''Silent Alarm'' * ''Luno the White Stallion ''Luno the White Stallion'' was a Terrytoons television series that aired in the mid-1960s. It centered on a little boy named Tim who had a toy horse Pegasus of marble white named Luno who would come alive and whisk him off on adventures in far of ...'', a Terrytoons television series * '' LUNO (band)'', a Czechoslovakian band * Loyola University New Orleans, a university in New Orleans * Bianco Luno (1795-1852), Danish entrepreneur {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squaring The Circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (\pi) is a transcendental number. That is, \pi is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if \pi were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Despite the proof that it is impossible, attempts to square the circle have be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the element of area of the surface, and is the line element along the boundary of . Here, is the Euler characteristic of . If the boundary is piecewise smooth, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crescent
A crescent shape (, ) is a symbol or emblem used to represent the lunar phase in the first quarter (the "sickle moon"), or by extension a symbol representing the Moon itself. In Hinduism, Lord Shiva is often shown wearing a crescent moon on his head symbolising that the lord is the master of time and is himself timeless. It is used as the astrological symbol for the Moon, and hence as the alchemical symbol for silver. It was also the emblem of Diana/Artemis, and hence represented virginity. In Christianity Marian veneration, it is associated with the Virgin Mary. From its use as roof finial in Ottoman era mosques, it has also become associated with Islam, and the crescent was introduced as chaplain badge for Muslim chaplains in the US military in 1993.On December 14, 1992, the Army Chief of Chaplains requested that an insignia be created for future Muslim chaplains, and the design (a crescent) was completed January 8, 1993. Emerson, William K., ''Encyclopedia of United St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that contains their diameters. The earliest known reference to this figure is in Archimedes's ''Book of Lemmas'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. Properties Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is Convex curve, convex, with diameter Area The area (geometry), area of the arbelos is equal to the area of a circle with diameter . Proof: For the proof, reflect the arbelos over the line through the points and , and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant (trigonometry)
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Chebotaryov
Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev, uk, Мико́ла Григо́рович Чеботарьо́в, russian: Никола́й Григо́рьевич Чеботарёв) ( – 2 July 1947) was a Ukrainian and Soviet mathematician. He is best known for the Chebotaryov density theorem. He was a student of Dmitry Grave, a Russian mathematician. Chebotaryov worked on the algebra of polynomials, in particular examining the distribution of the zeros. He also studied Galois theory and wrote a textbook on the subject titled ''Basic Galois Theory''. His ideas were used by Emil Artin to prove the Artin reciprocity law. He worked with his student Anatoly Dorodnov on a generalization of the quadrature of the lune, and proved the conjecture now known as the Chebotaryov theorem on roots of unity. Early life Nikolai Chebotaryov was born on 15 June 1894 in Kamianets-Podilskyi, Russian Empire (now in Ukraine). He entered the department of physics and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing. Early life Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians.Rothbard, MurrayDaniel Bernoulli and the Founding of Mathematical Economics ''Mises Institute'' (excerpted from ''An Austrian Perspective on the History of Economic Thought'') The Bernoulli family came originally from Antwerp, at that time in the Spanish Netherlands ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straightedge And Compass
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
