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Euclidean Symmetries
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations *Euclidean geometry, the study of the properties of Euclidean spaces *Non-Euclidean geometry, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines *Euclidean distance, the distance between pairs of points in Euclidean spaces *Euclidean ball, the set of points within some fixed distance from a center point Number theory *Euclidean division, the division which produces a quotient and a remainder *Euclidean algorithm, a method for finding greatest common divisors *Extended Euclidean algorithm, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b'' *Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus (mathematician), Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the philosophers Proclus and Pappus of Alexandria many centuries later. Until the early Renaissance he was often mistaken f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain. It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid (other)
Euclid, Euclides, or Eucleides generally refers to the ancient Greek mathematician Euclid of Alexandria (3rd century BC), who wrote a work on geometry called the ''Elements''. It may also refer to: People * Euclid of Megara (c. 435 BC–c. 365 BC), ancient Greek philosopher * Eucleides, archon of Athens (5th century BC) * Euclid Bertrand (born 1974), Dominican former footballer * Euclides da Cunha (1866–1909), Brazilian sociologist * Euclid Kyurdzidis (born 1968), Russian actor * Euclid Tsakalotos (born 1960), Greek economist and Minister of Finance * Nicholas Euclid (1932–2007), Australian rugby league player, coach, and official Mathematics, science, and technology * Euclid Contest, a maths competition held by the Centre for Education in Mathematics and Computing * Euclid (programming language) * Euclid (computer program) * Euclid, a computer system used by Euroclear * Euclid (spacecraft), a space telescope built by ESA, to be launched in Feb 2023 * Euclidean space * 4354 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Math League Of Eastern Massachusetts
The Intermediate Math League of Eastern Massachusetts (or IMLEM) is a math league for middle schools across Eastern Massachusetts. A brief history of IMLEM is given in its By-Laws: Schools As of 2017, 86 different schools attend the competition. Each school is allowed to send more than 1 team and each team can consist of at most 10 people. Alternates, people who are not officially part of team, can be taken too. There are a total of 15 different geographic clusters of schools and there is even a cluster of schools from Pennsylvania. The schools are then separated into different divisions with the schools in each division be approximately the same level. Schools can then make their way up through divisions to try to get into the top division, which is the Lexington Division. In total there are 13 divisions. Schools may send more than one team, however no student can compete on more than one team in a year. Also, a school may send alternates to gain the experience of a meet. Meet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Zoning
Zoning in the United States includes various land use laws falling under the police power rights of state governments and local governments to exercise authority over privately owned real property. Zoning laws in major cities originated with the Los Angeles zoning ordinances of 1904 and the New York City 1916 Zoning Resolution. Early zoning regulations were in some cases motivated by racism and classism, particularly with regard to those mandating single-family housing. Zoning ordinances did not allow African-Americans moving into or using residences that were occupied by majority whites due to the fact that their presence would decrease the value of home. The constitutionality of zoning ordinances was upheld by the Supreme Court of the United States in ''Village of Euclid, Ohio v. Ambler Realty Co.'' in 1926. According to the ''New York Times'', "single-family zoning is practically gospel in America," as a vast number of cities zone land extensively for detached single-family hom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance Map
A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into another representation, or it is simply endowed with an additional map or field. Distance fields can also be signed, in the case where it is important to distinguish whether the point is inside or outside of the shape. The map labels each pixel of the image with the distance to the nearest ''obstacle pixel''. A most common type of obstacle pixel is a ''boundary pixel'' in a binary image. See the image for an example of a Chebyshev distance transform on a binary image. Usually the transform/map is qualified with the chosen metric. For example, one may speak of Manhattan distance transform, if the underlying metric is Manhattan distance. Common metrics are: * Euclidean distance * Taxicab geometry, also known as ''City block distance'' or ''Manh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Relation
In mathematics, Euclidean relations are a class of binary relations that formalize " Axiom 1" in Euclid's ''Elements'': "Magnitudes which are equal to the same are equal to each other." Definition A binary relation ''R'' on a set ''X'' is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every ''a'', ''b'', ''c'' in ''X'', if ''a'' is related to ''b'' and ''c'', then ''b'' is related to ''c''.. To write this in predicate logic: :\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c). Dually, a relation ''R'' on ''X'' is left Euclidean if for every ''a'', ''b'', ''c'' in ''X'', if ''b'' is related to ''a'' and ''c'' is related to ''a'', then ''b'' is related to ''c'': :\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c). Properties # Due to the commutativity of ∧ in the definition's antecedent, ''aRb'' ∧ ''aRc'' even implies ''bRc'' ∧ ''cRb'' when ''R'' is right Euclidean. Similarly, ''bRa'' ∧ ''cRa'' implies ''bRc' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, . If the premise of the lemma does not hold, i.e., is a composite number, its consequent may be either true or false. For example, in the case of , , , composite number 10 divides , but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's Lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains. Formulations Euclid's lemma is commonly used in the following equivalent form: Euclid's lemma can be generalized as follows from prime numbers to any integers. This is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Euclidean Algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's identity, which are integers ''x'' and ''y'' such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are coprime. With that provision, ''x'' is the modular multiplicative inverse of ''a'' modulo ''b'', and ''y'' is the modular multiplicative inverse of ''b'' modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in Euclid's Elements, his ''Elements'' (c. 300 BC). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce Fraction (mathematics), fractions to their Irreducible fraction, simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
