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Zhao Youqin's π Algorithm
Zhao Youqin's algorithm was an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (, ? – 1330) to calculate the value of in his book ''Ge Xiang Xin Shu'' (). Algorithm Zhao Youqin started with an inscribed square in a circle with radius r. If \ell denotes the length of a side of the square, draw a perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ... line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram: :d=\sqrt :e=r-d=r-\sqrt. Extend the perpendicular line d to dissect the circle into an octagon; \ell_2 denotes the length of one side of octagon. :\ell_2=\sqrt :\ell_2=\frac\sqrt Let l_3 denotes the length of a side of hexadecagon :\ell_3=\frac\sqrt similarly :\ell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zhao Youqin Circle Dissection Algorithm
Zhao may refer to: * Zhao (surname) (赵), a Chinese surname ** commonly spelled Chao in Taiwan or up until the early 20th century in other regions ** Chiu, from the Cantonese pronunciation ** Cho (Korean surname), represent the Hanja 趙 (Chinese: Zhao) ** Triệu, a Vietnamese surname which is the equivalent of the Mandarin Chinese surname Zhao (趙) * Zhao County, in Shijiazhuang, Hebei, China * Zhao family (other) ** Zhao family (Internet slang), based on the surname Zhao, an internet term in China which refers to the ruling elite and the rich * 兆 (zhào), a Chinese numeral which usually represents 106 or 1012 **Mega-, corresponding SI prefix in China, equals to 106 **Tera-, corresponding SI prefix in Taiwan, equals to 1012 * Admiral Zhao, a character in the animated series ''Avatar: The Last Airbender'' Chinese history * Zhao (state) (403 BC–222 BC), a Warring States period state * Triệu dynasty (204 BC–111 BC), or Zhao dynasty, the ruling house of the Nan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuan Dynasty
The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fifth khagan-emperor of the Mongol Empire from the Borjigin clan, and lasted from 1271 to 1368. In orthodox Chinese historiography, the Yuan dynasty followed the Song dynasty and preceded the Ming dynasty. Although Genghis Khan had been enthroned with the Han-style title of Emperor in 1206 and the Mongol Empire had ruled territories including modern-day northern China for decades, it was not until 1271 that Kublai Khan officially proclaimed the dynasty in the traditional Han style, and the conquest was not complete until 1279 when the Southern Song dynasty was defeated in the Battle of Yamen. His realm was, by this point, isolated from the other Mongol-led khanates and controlled most of modern-day China and its surrounding areas, including ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the spoke of a chariot wheel. as a function of axial position ../nowiki>" Spherical coordinates In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane. See also *Bend radius *Filling radius in Riemannian geometry *Radius of convergence * Radius of convexity *Radius of curvature *Radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoshio Mikami
was a Japanese mathematician and historian of ''Japanese mathematics''. He was born February 16, 1875, in Kotachi, Hiroshima prefecture. He attended the High School of Tohoku University, and in 1911 was admitted to the Imperial University of Tokyo. He studied history of Japanese and Chinese mathematics. In 1913, he published "The Development of Mathematics in China and Japan" in Leipzig.Yoshio Mikami, The Development of Mathematics in China and Japan, 1913, Library of Congress 61-13497 This book consisted of two parts with 47 chapters. Part one has 21 chapters that describe in depth several important Chinese mathematicians and mathematical classics including Liu Hui, Shen Kuo, Qin Jiushao, Sun Tzu, The Nine Chapters on the Mathematical Art, Mathematical Treatise in Nine Sections, Li Ye, Zhu Shijie and study on π. Part II deals with important ''wasan'' mathematicians and their works, including Kambei Mori, Yoshida Koyu, Kowa Seki, Imamura Chisho, Takahara Kisshu, Kurushima, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. Properties of the general octagon The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).Dao Thanh Oai (2015), "Equilatera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexadecagon
In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. Regular hexadecagon A '' regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is and can be constructed as a truncated octagon, t, and a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon, . Construction As 16 = 24 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians. Measurements Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees. The area of a regular hexadecagon with edge length ''t'' is :\begin A = 4t^2 \cot \frac =& 4t^2 \left(1+\sqrt+\sqrt\right)\\ =& 4t^2 (\sqrt+1)(\sqrt+1) .\end Because the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius ''R'' by truncatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liu Hui's π Algorithm
Liu Hui's algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of the state of Cao Wei. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while Zhang Heng (78–139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, ) or as \pi \approx \sqrt \approx 3.162. Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician Wang Fan (219–257) provided . All these empirical values were accurate to two digits (i.e. one decimal place). Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: . Liu Hui remarked in his commentary to ''The Nine Chapters on the Mathematical Art'', that the ratio of the circumference of an inscribed hexagon to the diamete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chinese Mathematics
Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geometry, number theory and trigonometry. Since the Han Dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like continued fractions are widely used and have been well-documented ever-since. They deliberately find the principal ''n''th root of positive numbers and the roots of equations. The major texts from the period, ''The Nine Chapters on the Mathematical Art'' and the ''Book on Numbers and Computation'' gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
