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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 independent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotating Hexlet Equator Opt
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions. If the rotation axis passes internally through the body's own center of mass, then the body is said to be ''autorotating'' or '' spinning'', and the surface intersection of the axis can be called a ''pole''. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called ''revolving'' or ''orbiting'', typically when it is produced by gravity, and the ends of the rotation axis can be called the ''orbital poles''. Mathematics Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (mathematics), eccentricity e, a number ranging from e = 0 (the Limiting case (mathematics), limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytic geometry, Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scripta Mathematica
''Scripta Mathematica'' was a quarterly journal published by Yeshiva University devoted to the philosophy, history, and expository treatment of mathematics. It was said to be, at its time, "the only mathematical magazine in the world edited by specialists for laymen.". The journal was established in 1932 under the editorship of Jekuthiel Ginsburg, a professor of mathematics at Yeshiva University, and its first issue appeared in 1933 at a subscription price of three dollars per year. It ceased publication in 1973. Notable papers published in ''Scripta Mathematica'' included work by Nobelist Percy Williams Bridgman concerning the implications for physics of set-theoretic paradoxes, and Hermann Weyl's obituary of Emmy Noether Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe .... Some s ... [...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] |
Cun (unit)
A ''cun'' (), often glossed as the ''Chinese inch'', is a traditional Chinese unit of length. Its traditional measure is the width of a person's thumb at the knuckle, whereas the width of the two forefingers denotes 1.5 cun and the width of four fingers (except the thumb) side-by-side is 3 cuns. It continues to be used to chart acupuncture points on the human body, and, in various uses for traditional Chinese medicine. The cun was part of a larger decimal system. A cun was made up of 10 fen, which depending on the period approximated lengths or widths of millet grains, and represented one-tenth of a chi ("Chinese foot"). In time the lengths were standardized, although to different values in different jurisdictions. (See chi (unit) for details.) In Hong Kong, using the traditional standard, it measures ~3.715 cm (~1.463 in) and is written "tsun". In the twentieth century in the Republic of China, the lengths were standardized to fit with the metric system, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samukawa Shrine
is a Shinto shrine in the Miyayama neighborhood of the town of Samukawa, Kōza District. Kanagawa Prefecture, Japan. It is the ''ichinomiya'' of former Sagami Province. The main festival of the shrine is held annually on September 20. This shrine is one of the most famous shrines around Tokyo, where about 2 million people visit each year. Enshrined ''kami'' The ''kami'' enshrined at Samukawa Jinja is: * , an amalgamation of the male and the female Beppyo shrines History The origins of Samukawa Shrine are unknown. Unverifiable shrine legend states that during the reign of Emperor Yūryaku (418-479), messengers were sent to this shrine from the imperial court. The earliest written records indicate that the shrine was rebuilt in the year 727, and its name also appears in the ''Shoku Nihon Kōki'' entry for the year 846. By the time of the 923 AD ''Engishiki'', the shrine is styled as the only shrine in Sagami Province to be a . There is also a mystery regarding the ''kami'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sangaku
Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes. History The Sangaku were painted in color on wooden tablets ( 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 and foreign relations for ... [...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 The Japanese mathematical Model (abstract), schema evolved during a period when Japan's people were isolated from European influences, but instead borrowed from ancient mathematical texts written in China, including those from the Yuan dynasty and earlier. The Japanese mathematicians Yoshida Koyu, Yoshida Shichibei Kōyū, Imamura Chishō, and T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sangaku Of Soddy's Hexlet In Samukawa Shrine
Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes. History The Sangaku were painted in color on wooden tablets ( 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 and foreign relations for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
