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Magic Square
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The "order" of the magic square is the number of integers along one side (''n''), and the constant sum is called the "magic constant". If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square". Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the #Sagrada Família magic square, Sagrada Família magic square and the #Parker square, Parker square, are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pandiagonal Magic Square
A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant. A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n \times n pandiagonal magic square can be regarded as having 8n^2 orientations. 3×3 pandiagonal magic squares It can be shown that non-trivial pandiagonal magic squares of order 3 do not exist. Suppose the square :\begin \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \end is pandiagonally magic with magic constant . Adding sums and results in . Subtract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of The Brethren Of Purity
The ''Encyclopedia of the Brethren of Purity'' (, Rasā'il Ikhwān al-ṣafā') also variously known as the ''Epistles of the Brethren of Sincerity'', ''Epistles of the Brethren of Purity'' and ''Epistles of the Brethren of Purity and Loyal Friends'' is an Islamic encyclopedia"The work only professes to be an epitome, an outline; its authors lay claim to no originality, they only summarize what others have thought and discovered. What they do lay claim to is system and completeness. The work does profess to contain a ''systematized'', harmonious and co-ordinated view of the universe and life, its origin and destiny, formed out of many discordant, incoherent views; and it does claim to be a 'complete account of all things' - to contain, in epitome, all that was known at the time it was written. It refers to more profound and special treatises for fuller information on the several sciences it touches upon, but it does claim to touch on all sciences, all departments of knowledge, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Most-perfect Magic Square
A most-perfect magic square of order ''n'' is a magic square containing the numbers 1 to ''n''2 with two additional properties: # Each 2 × 2 subsquare sums to 2''s'', where . # All pairs of integers distant ''n''/2 along a (major) diagonal sum to ''s''. There are 384 such combinations. Examples Two 12 × 12 most-perfect magic squares can be obtained adding 1 to each element of: [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [1,] 64 92 81 94 48 77 67 63 50 61 83 78 [2,] 31 99 14 97 47 114 28 128 45 130 12 113 [3,] 24 132 41 134 8 117 27 103 10 101 43 118 [4,] 23 107 6 105 39 122 20 136 37 138 4 121 [5,] 16 140 33 142 0 125 19 111 2 109 35 126 [6,] 75 55 58 53 91 70 72 84 89 86 56 69 [7,] 76 80 93 82 60 65 79 51 62 49 95 66 [8,] 115 15 98 13 131 30 112 4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang Hui
Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's triangle. This triangle was the same as Pascal's triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary of Qin Jiushao, another well-known Chinese mathematician. Written work The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's book ''Xiángjiě Jiǔzhāng Suànfǎ '' () of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's triangle" was invented by mathematician Jia XianNeedham, Volume 3, 134-137. who expounded it around 1100 AD, about 500 years before Pascal. His book (now lost), known as ''Rújī Shìsuǒ'' () or ''Piling-up Powers and Unlocking Coeffici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cheng Dawei
Cheng Dawei (程大位, 1533–1606), also known as Da Wei Cheng or Ch'eng Ta-wei, was a Chinese mathematician and writer who was known mainly as the author of '' Suanfa Tongzong (算法統宗)'' (''General Source of Computational Methods''). He has been described as "the most illustrious Chinese arithmetician." Almost all that is known about his life is contained in a passage written in the Preface of the book by one of his descendants when the book was being reprinted: :In his youth my ancestor Cheng Da Wei was academically gifted, but although he was well versed in scholarly matters, he continued to carry out his profession as a sincere Local Agent, without becoming a scholar. He never lagged behind either on the classics or on ancient writings with old style characters, but was particularly gifted in arithmetic. In the prime of his life he visited the fairs of Wu and Chu. When he came across books that talked about "square fields" or "grain with the husk removed" ... he never ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suanfa Tongzong
''Suanfa tongzong'' ( zh, 算法統宗) is a mathematical text written by sixteenth century Chinese mathematician Cheng Dawei (1533–1606) and published in the year 1592. The book contains 595 problems divided into 17 chapters. The book is essentially general arithmetic for the abacus. The book was the main source available to scholars concerning mathematics as it developed in China's tradition. Six years after the publication of Suanfa Tongzong, Cheng Dawei published another book titled ''Suanfa Zuanyao'' (''A Compendium of calculating Methods''). About 90% of the content of the new book came from the contents of four chapters of the first book with some rearrangement. It is said that when Suanfa Tongzong was first published, it sold so many copies that the cost of paper went up and the lucrative sales resulted in unscrupulous people beginning to print pirated copies of the book with many errors. It was this that forced the author to print an abridged version. Some features ''Su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuel Moschopoulos
Manuel Moschopoulos ( Latinized as Manuel Moschopulus; ), was a Byzantine commentator and grammarian, who lived during the end of the 13th and the beginning of the 14th century and was an important figure in the Palaiologan Renaissance. ''Moschopoulos'' means "little calf," and is probably a nickname. Life Moschopoulos was a student of Maximos Planudes and possibly his successor as a head of a school in Constantinople, where he taught throughout his life. A mysterious and ill-documented excursion into politics led to his imprisonment for a while. Works His chief work is ''Erotemata grammaticalia'' (), in the form of question and answer, based upon an anonymous epitome of grammar, and supplemented by a lexicon of Attic nouns. He was also the author of ''scholia'' on the first and second books of the ''Iliad'', on Hesiod, Theocritus, Pindar and other classical and later authors; of riddles, letters, and a treatise on the magic squares. His grammatical treatises formed the foundati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuan Dynasty
The Yuan dynasty ( ; zh, c=元朝, p=Yuáncháo), officially the Great Yuan (; Mongolian language, Mongolian: , , literally 'Great Yuan State'), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after Division of the Mongol Empire, its division. It was established by Kublai (Emperor Shizu or Setsen Khan), the fifth khagan-emperor of the Mongol Empire from the Borjigin clan, and lasted from 1271 to 1368. In Chinese history, the Yuan dynasty followed the Song dynasty and preceded the Ming dynasty. Although Genghis Khan's enthronement as Khagan in 1206 was described in Chinese language, Chinese as the Han Chinese, Han-style title of Emperor of China, Emperor 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 t ... [...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] |
