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Problem Of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonius Problem Typical Solution
Apollonius ( grc, Απολλώνιος) is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apollonius (son of Archias), sculptor Historians * Apollonius of Aphrodisias (fl. c. 3rd century BC), historian of Caria * Apollonius of Ascalon, historian mentioned by Stephanus of Byzantium Writers * Apollonius Attaleus, writer on dreams * Apollonius of Acharnae, ancient Greek writer on festivals * Apollonius of Laodicea, writer on astrology * Apollonius of Rhodes (born c. 270 BC), librarian and poet, best known for the ''Argonautica'' * Apollonius (son of Chaeris), ancient Greek writer, mentioned by the scholiast on Aristophanes * Apollonius (son of Sotades), writer Oratory * Apollonius Dyscolus (fl. 2nd century AD), grammarian * Apollonius Eidographus, ancient Greek grammarian * Apollonius Molon (fl. 70 BC), rh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Diaz Gergonne
Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion because the French government feared a foreign invasion intended to undo the French Revolution and restore Louis XVI to the throne of France. He saw action in the major battle of Valmy on 20 September 1792. He then returned to civilian life but soon was called up again and took part in the French invasion of Spain in 1794. In 1795, Gergonne and his regiment were sent to Nîmes. At this point, he made a definitive transition to civilian life by taking up the chair of "transcendental mathematics" at the new École centrale. He came under the influence of Gaspard Monge, the Director of the new École polytechnique in Paris. In 1810, in response to difficulties he encountered in trying to publish his work, Gergonne founded his own mathematics jour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the Roman Republic it became the dominant language in the Italian region and subsequently throughout the Roman Empire. Even after the fall of Western Rome, Latin remained the common language of international communication, science, scholarship and academia in Europe until well into the 18th century, when other regional vernaculars (including its own descendants, the Romance languages) supplanted it in common academic and political usage, and it eventually became a dead language in the modern linguistic definition. Latin is a highly inflected language, with three distinct genders (masculine, feminine, and neuter), six or seven noun cases (nominative, accusative, genitive, dative, ablative, and vocative), five declensions, four verb conjuga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circles Of Apollonius
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection. The main uses of this term are fivefold: # Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ''ratio'' of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2. # The Apollonian circles are two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Circle Method
In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem. History The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines. Hundreds of papers followed, and the method still yields results. The method is the subject of a monograph by R. C. Vaughan. Outline The goal is to prove asymptotic behavior of a series: to show that for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). Technically, the genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ford Circles
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles are either tangent or disjoint from each other. History Ford circles are a special case of mutually tangent circles; the base line can be thought of as a circle with infinite radius. Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius and the Apollonian gasket are named.. In the 17th century René Descartes discovered Descartes' theorem, a relationship between the reciprocals of the radii of mutually tangent circles. Ford circles also appear in the Sangaku (geometrical puzzles) of Japanese mathematics. A typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. Construction The construction of the Apollonian gasket starts with three circles C_1, C_2, and C_3 (black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles C_4 and C_5 (red) that are tangent to all three of the original circles – these are called ''Apollonian circles''. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descartes' Theorem
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643. History Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic, ''De tactionibus'' 'On tangencies'' It has been lost, and is known only through mentions of it in other works. René Descartes discussed the problem briefly in 1643, in a letter to Princess Elisabeth of the Palatinate. He came up with the equation describing the relation between the radii, or curvatures, of four pairwise tangent circles. This result became known as Descartes' theorem. This result was rediscovered in 1826 by Jakob Steiner, in 1842 by Philip Beecr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, later becoming a central intellectual of the Dutch Golden Age. Although he served a Protestant state and was later counted as a deist by critics, Descartes considered himself a devout Catholic. Many elements of Descartes' philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
