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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 ...
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Sangaku Of Konnoh Hachimangu 1859
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 ...
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Sangaku Three Circles
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 ...
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Sangaku At Enmanji
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 ...
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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 independen ...
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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 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 Shichibei Kōyū, Imamura Chishō, and Takahara Kisshu are among the ...
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Equal Incircles Theorem
In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal. In the illustration the equal blue circles define the spacing between the rays, as described. The theorem states that the incircles of the triangles formed (starting from any given ray) by every other ray, every third ray, etc. and the base line are also equal. The case of every other ray is illustrated above by the green circles, which are all equal. From the fact that the theorem does not depend on the angle of the initial ray, it can be seen that the theorem properly belongs to analysis, rather than geometry, and must relate to a continuous scaling function which defines the spacing of the rays. In fact, this function is the hyperbolic sine. The theorem is a direct corollary of the following lemma: ...
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Japanese Theorem For Concyclic Polygons
__notoc__ In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The Japanese theorem follows from Carnot's theorem; it is a Sangaku problem. Proof This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic ''quadrilateral'', the sum of inradii of triangles is constant. After proving the quadrilateral case, the general case of the cyclic polygon theorem is an immediate corollary. The quadrilateral rule can be applied to quadrilateral components of a general partition of a cyclic polygon, and repeated application of the rule, which "flips" one diagonal, will generate all the possible partitions from any given partition, with each "flip" preserving the sum of the inradii. ...
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Japanese Theorem For Concyclic Quadrilaterals
In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle. Specifically, let be an arbitrary cyclic quadrilateral and let , , , be the incenters of the triangles , , , . Then the quadrilateral formed by , , , is a rectangle. Note that this theorem is easily extended to prove the Japanese theorem for cyclic polygons. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral. The construction shows that the parallelogram is a rhombus, which is equivalent to showing that the sums of the radii of the incircles tangent to each diagonal are equal. The quadrilateral ca ...
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Seki Takakazu
, Selin, Helaine. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890 also known as ,Selin, was a Japanese mathematician and author of the Edo period. Seki laid foundations for the subsequent development of Japanese mathematics, known as ''wasan''. He has been described as "Japan's Newton". He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. Although he was a contemporary of German polymath mathematician and philosopher Gottfried Leibniz and British polymath physicist and mathematician Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo period. While it is not clear how much of the achievements of ''wasan'' are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered i ...
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Daniel Pedoe
Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have remained in print for decades and been translated into several languages. These books include the three-volume ''Methods of Algebraic Geometry'' (which he wrote in collaboration with W. V. D. Hodge), ''The Gentle Art of Mathematics'', ''Circles: A Mathematical View'', ''Geometry and the Visual Arts'' and most recently ''Japanese Temple Geometry Problems: San Gaku'' (with Hidetoshi Fukagawa). Early life Daniel Pedoe was born in London in 1910, the youngest of thirteen children of Szmul Abramski, a Jewish immigrant from Poland who found himself in London in the 1890s: he had boarded a cattleboat not knowing whether it was ...
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Ema (Shinto)
are small wooden plaques, common to Japan, in which Shinto and Buddhist worshippers write prayers or wishes. are left hanging up at the shrine, where the (spirits or gods) are believed to receive them. Typically wide and tall, they often carry images or are shaped like animals, or symbols from the zodiac, Shinto, or the particular shrine or temple. In ancient times, people would donate horses to the shrines for good favor; over time this was transferred to a wooden plaque with a picture of a horse, and later still to the various wooden plaques sold today for the same purpose. Once inscribed with a wish, are hung at the shrine until they are ritually burned at special events, symbolic of the liberation of the wish from the writer. History In some early Shinto and folk traditions of Japan, horses were seen as carrying messages from the , and were usually used to transmit requests during droughts or famines. Horses were extremely expensive, and figures made of clay or wood ...
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Ford Circle
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 two ...
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