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Point Groups In Four Dimensions
In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere. History on four-dimensional groups * 1889 Édouard Goursat, ''Sur les substitutions orthogonales et les divisions régulières de l'espace'', Annales Scientifiques de l'École Normale Supérieure, Sér. 3, 6, (pp. 9–102, pp. 80–81 tetrahedra), Goursat tetrahedron * 1951, A. C. Hurley, ''Finite rotation groups and crystal classes in four dimensions'', Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650 * 1962 A. L. MacKay ''Bravais Lattices in Four-dimensional Space'' * 1964 Patrick du Val, ''Homographies, quaternions and rotations'', quaternion-based 4D point groups * 1975 Jan Mozrzymas, Andrzej Solecki, ''R4 point groups'', Reports on Mathematical Physics, Volume 7, Issue 3, p. 363-394 * 1978 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polychoral Group Tree
An antiphon (Greek ἀντίφωνον, ἀντί "opposite" and φωνή "voice") is a short chant in Christian ritual, sung as a refrain. The texts of antiphons are usually taken from the Psalms or Scripture, but may also be freely composed. Their form was favored by St Ambrose and they feature prominently in Ambrosian chant, but they are used widely in Gregorian chant as well. They may be used during Mass, for the Introit, the Offertory or the Communion. They may also be used in the Liturgy of the Hours, typically for Lauds or Vespers. They should not be confused with Marian antiphons or processional antiphons. When a chant consists of alternating verses (usually sung by a cantor) and responses (usually sung by the congregation), a refrain is needed. The looser term antiphony is generally used for any call and response style of singing, such as the kirtan or the sea shanty and other work songs, and songs and worship in African and African-American culture. Antiphonal musi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational symmetry is the number of distinct Orientation (geometry), orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids. Formal treatment Formally the rotational symmetry is symmetry with respect to some or all rotations in -dimensional Euclidean space. Rotations are Euclidean group#Direct and indirect isometries, direct isometries, i.e., Isometry, isometries preserving Orientation (mathematics), orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of (see Euclidean g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Symmetry
In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation. An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetries is called ''centrosymmetric''. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Inversion
In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point (geometry), point is reflected across a designated inversion center, which remains Fixed point (mathematics), fixed. In Euclidean space, Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves Euclidean distance, distance). In the Euclidean plane, a point reflection is the same as a half turn, half-turn rotation (180° or radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but orientation-reversing, reverses orientation. A point reflection is an involution (mathematics), involution: applying it twice is the identity transformation. An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter's Notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. Reflectional groups For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors. The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the ''A''''n'' group is represented by ''n''−1 to imply ''n'' nodes connected by ''n−1'' order-3 branches. Exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford
Oxford () is a City status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the List of oldest universities in continuous operation, oldest university in the English-speaking world; it has buildings in every style of Architecture of England, English architecture since late History of Anglo-Saxon England, Anglo-Saxon. Oxford's industries include motor manufacturing, education, publishing, science, and information technologies. Founded in the 8th century, it was granted city status in 1542. The city is located at the confluence of the rivers Thames (locally known as the Isis) and River Cherwell, Cherwell. It had a population of in . It is north-west of London, south-east of Birmingham and north-east of Bristol. History The history of Oxford in England dates back to its original settlement in the History of Anglo-Saxon England, Saxon period. The name � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including ''The Fifty-Nine Icosahedra'' (1938) and ''Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Number
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number of an irreducible root system. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is where is the rank, and is the number of reflections. In the crystallographic case, is half the number of roots; and is the dimension of the corresponding semisimple Lie algebra. *If th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
