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Soddy Circle
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. Frederick Soddy's 1936 poem ''The Kiss Precise'' summarizes the theorem in terms of the ''bends'' (signed inverse radii) of the four circles: Special cases of the theorem apply when one or two of the circles is replaced by a straight line (with zero bend) or when the bends are integers or square numbers. A version of the theorem using complex numbers allows the centers of the circles, and not just their radii, to be calculated. With an appropriate definition of curvature, the theorem also applies in spherical geometry and hyperbolic geometry. In higher dimensions, an analogous quadratic equation applies to systems of pairwise tangent spheres or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversive Geometry
In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces. Inversion in a circle Inverse of a point To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point ''P'' with respect to a ''reference circle (Ø)'' with center ''O'' and radius ''r'' is a point ''P'', lying on the ray from ''O'' through ''P'' such that :OP \cdot OP^ = r^2. This is calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viviani's Theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world. Proof This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA 2010, , p. 96 () Let ABC be an equilateral triangle whose height is ''h'' and whose side is ''a''. Let P be any point inside the triangle, and ''s, t, u'' the perpendicular distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA. Now, the areas of these tria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Chain
In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction The arbelos is defined by two circles, and , which are tangent at the point and where is enclosed by . Let the radii of these two circles be denoted as , respectively, and let their respective centers be the points . The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to (the inner circle) and internally tangent to (the outer circle). Let the radius, diameter and center point of the th circle of the Pappus chain be denoted as , respectively. Properties Centers of the circles Ellipse All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the th circle of the Pappus chain to the two centers of the arbelos circles equals a constant \overline + \overline = (r_U + r_n) + (r_V - r_n) = r_U + r_V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and for his generalization of Descartes' theorem on tangent circles to four and higher dimensions. Biography Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs,UK Census 1871, RG10-863-89-23 and his wife Eleanor Gosset (formerly Thorold). He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. In 1900 he married Emily Florence Wood, and they subsequently had two children, named Kathleen and John.UK Census 1911, RG14-181-9123-19 Mathematics According to H. S. M. Coxeter, after obtaining his law degree in 1896 and having no client ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soddy's Hexlet
In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the red inner sphere and two spheres (not shown) above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere (the blue outer sphere in Figure 1), which is not tangent to the three others. According to a theorem published by Frederick Soddy in 1937, it is always possible to find a hexlet for any choice of mutually tangent spheres ''A'', ''B'' and ''C''. Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres (Figure 1); in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nature (journal)
''Nature'' is a British weekly scientific journal founded and based in London, England. As a multidisciplinary publication, ''Nature'' features Peer review, peer-reviewed research from a variety of academic disciplines, mainly in science and technology. It has core editorial offices across the United States, continental Europe, and Asia under the international scientific publishing company Springer Nature. ''Nature'' was one of the world's most cited scientific journals by the Science Edition of the 2022 ''Journal Citation Reports'' (with an ascribed impact factor of 50.5), making it one of the world's most-read and most prestigious academic journals. , it claimed an online readership of about three million unique readers per month. Founded in the autumn of 1869, ''Nature'' was first circulated by Norman Lockyer and Alexander MacMillan (publisher), Alexander MacMillan as a public forum for scientific innovations. The mid-20th century facilitated an editorial expansion for the j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous '' Journal'' (1826). After Steiner's publication (1832) of his ''Systematische Entwickelungen'' he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834). This he occupied until his death in Bern on 1 April 1863. He was described by Thomas Hirst as follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sangaku
Sangaku or san gaku () are Japanese Euclidean geometry, geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples in Japan, Buddhist temples during the Edo period by members of all social classes. History The sangaku were painted in color on wooden tablets (Ema (Shinto), ema) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas, as challenges to the congregants, or as displays of the solutions to questions. Many of these tablets were lost during the period of modernization that followed the Edo period, but around nine hundred are known to remain. Fujita Kagen (1765–1821), a Japanese mathematician of prominence, published the first collection of ''sangaku'' problems, his ''Shimpeki Sampo'' (Mathematical problems Suspended from the Temple) in 1790, and in 1806 a sequel, the ''Zoku Shimpeki Sampo''. During this period Japan applied strict regulations to commerce a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yamaji Nushizumi
, was a Japanese mathematician in the middle of the Edo period, focused on calendar reform. Life The Yamaji family claimed to be descendants of Taira no Shigemori, and were vassals of the shogunate ever since an ancestor Yamaji Souemon Hisanaga was employed as an attendant. In the 9th year of Kyoho (1724), when Nushizumi was 21 years old, he started working as a servant. After that, he became an accountant in 1738, but was dismissed as "not good at work", and in 1739, he entered the service. In the first year of Kanen (1748), he became a provisional helper for the supplementary calendar. He served as an assistant to Shibukawa Noriyoshi and Nishikawa Masayoshi, an astronomical director, and traveled between Kyoto and Edo. As a close aide of Tokugawa Yoshimune his teacher Nakane planned to convert the calendar to the Gregorian calendar, and he participated in this project. But the conversion to the Western calendar was not taken up, and as a result the Horeki calendar was only a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Japanese Mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese mathematical theory from Western mathematics (洋算 ''yōsan''). In the history of mathematics, the development of ''wasan'' falls outside the Western realm. At the beginning of the Meiji period (1868–1912), Japan and its people opened themselves to the West. Japanese scholars adopted Western mathematical technique, and this led to a decline of interest in the ideas used in ''wasan''. History Pre-Edo period (552-1600) Records of mathematics in the early periods of Japanese history are nearly nonexistent. Though it was at this time that a large influx of knowledge from China reached Japan, including that of reading and writing, little sources exist of usage of mathematics within Japan. However, it is suggested that this period saw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
