Associative Magic Square
An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an ''n'' × ''n'' square, filled with the numbers from 1 to ''n''2, this common sum must equal ''n''2 + 1. These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares. Examples For instance, the Lo Shu Square – the unique 3 × 3 magic square – is associative, because each pair of opposite points form a line of the square together with the center point, so the sum of the two opposite points equals the sum of a line minus the value of the center point regardless of which two opposite points are chosen. The 4 × 4 magic square from Albrecht Dürer 1514 engraving – also found in a 1765 letter of Benjamin Franklin – is also associative, with each pair of opposite numbers summing to 17. Existence and enumeration The num ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Magic Square Lo Shu
Magic or magick most commonly refers to: * Magic (supernatural), beliefs and actions employed to influence supernatural beings and forces ** ''Magick'' (with ''-ck'') can specifically refer to ceremonial magic * Magic (illusion), also known as stage magic, the art of appearing to perform supernatural feats * Magical thinking, the belief that unrelated events are causally connected, particularly as a result of supernatural effects Magic or magick may also refer to: Art and entertainment Film and television * ''Magic'' (1917 film), a silent Hungarian drama * ''Magic'' (1978 film), an American horror film * ''Magic'', a 1983 Taiwanese film starring Wen Chao-yu * Magic (TV channel), a British music television station Literature * Magic in fiction, the genre of fiction that uses supernatural elements as a theme * '' Magic: A Fantastic Comedy'', a 1913 play by G. K. Chesterton * ''Magic'' (short story collection), a 1996 short story collection by Isaac Asimov * ''Magic'' ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Albrecht Dürer - Melencolia I (detail)
Albrecht ("noble", "bright") is a given name or surname of German origin and may refer to: First name * Albrecht Agthe, (1790–1873), German music teacher * Albrecht Altdorfer, (c. 1480–1538) German Renaissance painter * Albrecht Becker, (1906–2002), German production designer, photographer, and actor * Albrecht Berblinger, (1770–1829), German constructor (the tailor of ulm) * Albrecht Brandi, (1914–1966), German U-boat commander in World War II * Archduke Albrecht, Duke of Teschen (1817–1895) general who controlled the Austrian Army * Albrecht, Duke of Württemberg, (1865–1939), German field marshal in World War I * Albrecht von Wallenstein, (1583–1634), Bohemian soldier and politician during the Thirty Years' War * Albrecht Dieterich, (1866–1908) German classical philologist and religious scholar * Albrecht Dietz, (1926–2012), German entrepreneur and scientist * Albrecht Dürer, (1471–1528), German artist and mathematician * Albrecht Dürer the Elder, Ge ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Lo Shu Square
The Luoshu (pinyin), Lo Shu (Wade-Giles), or Nine Halls Diagram is an Ancient China, ancient Chinese diagram and named for the Luo River (Henan), Luo River near Luoyang, Henan. The Luoshu appears in Chinese mythology, myths concerning the Chinese inventions, invention of Chinese writing, writing by Cangjie and other culture heroes. It is a unique normal magic square of order three. It is usually paired with the Yellow River Map, River Map or Hetunamed in reference to the Yellow Riverand used with the River Map in various contexts involving Chinese geomancy, Chinese numerology, numerology, Chinese philosophy, philosophy, and early natural science. Traditions The Lo Shu is part of the legacy of ancient Chinese mathematical and divination (cf. the I Ching ) traditions, and is an important emblem in ''Feng Shui'' ()—the art of geomancy concerned with the placement of objects in relation to the flow of qi (), or "natural energy". History A Chinese legend concerning the pre-histori ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Albrecht Dürer
Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prints, printmaker, and history of geometry#Renaissance, theorist of the German Renaissance. Born in Free Imperial City of Nuremberg, Nuremberg, Dürer established his reputation and influence across Europe in his twenties due to his high-quality List of woodcuts by Dürer, woodcut prints. He was in contact with the major Italian artists of his time, including Raphael, Giovanni Bellini and Leonardo da Vinci, and from 1512 was patronized by Holy Roman Emperor, Emperor Maximilian I, Holy Roman Emperor, Maximilian I. Dürer's vast body of work includes List of engravings by Dürer, engravings, his preferred technique in his later prints, Altarpiece, altarpieces, portraits and self-portraits, watercolours and books. The woodcuts series are stylist ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Melencolia I
''Melencolia I'' is a large 1514 engraving by the German Renaissance artist Albrecht Dürer. Its central subject is an enigmatic and gloomy winged female figure thought to be a personification of melancholia – melancholy. Holding her head in her hand, she stares past the busy scene in front of her. The area is strewn with symbols and tools associated with craft and carpentry, including an hourglass, weighing scales, a hand plane, a claw hammer, and a saw. Other objects relate to alchemy, geometry or numerology. Behind the figure is a structure with an embedded magic square, and a ladder leading beyond the frame. The sky contains a rainbow, a comet or planet, and a bat-like creature bearing the text that has become the print's title. Dürer's engraving is one of the most well-known extant old master prints, but, despite a vast art-historical literature, it has resisted any definitive interpretation. Dürer may have associated melancholia with creative activity; the woman may ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Benjamin Franklin
Benjamin Franklin (April 17, 1790) was an American polymath: a writer, scientist, inventor, statesman, diplomat, printer, publisher and Political philosophy, political philosopher.#britannica, Encyclopædia Britannica, Wood, 2021 Among the most influential intellectuals of his time, Franklin was one of the Founding Fathers of the United States; a Committee of Five, drafter and signer of the United States Declaration of Independence, Declaration of Independence; and the first United States Postmaster General, postmaster general. Born in the Province of Massachusetts Bay, Franklin became a successful Early American publishers and printers, newspaper editor and printer in Philadelphia, the leading city in the colonies, publishing ''The Pennsylvania Gazette'' at age 23. He became wealthy publishing this and ''Poor Richard's Almanack'', which he wrote under the pseudonym "Richard Saunders". After 1767, he was associated with the ''Pennsylvania Chronicle'', a newspaper known for it ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Singly Even
In mathematics an even integer, that is, a number that is divisibility, divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. Specifically, the 2-order of a nonzero integer ''n'' is the maximum integer value ''k'' such that ''n''/2''k'' is an integer. This is equivalent to the multiplicity (mathematics), multiplicity of 2 in the prime factorization. *A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd. *A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Singular Matrix
A singular matrix is a square matrix that is not invertible, unlike non-singular matrix which is invertible. Equivalently, an n-by-n matrix A is singular if and only if determinant, det(A)=0. In classical linear algebra, a matrix is called ''non-singular'' (or invertible) when it has an inverse; by definition, a matrix that fails this criterion is singular. In more algebraic terms, an n-by-n matrix A is singular exactly when its columns (and rows) are linearly dependent, so that the linear map x\rightarrow Ax is not one-to-one. In this case the kernel ( null space) of A is non-trivial (has dimension ≥1), and the homogeneous system Ax = 0 admits non-zero solutions. These characterizations follow from standard rank-nullity and invertibility theorems: for a square matrix A, det(A) \neq 0 if and only if rank(A)= n, and det(A) = 0 if and only if rank(A)3 then it is a singular matrix. * Numerical noise/ Round off: In numerical computations, a matrix may be nearly singular when its ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |