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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 ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 )
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Mathematics
MATHEMATICS (from Greek μάθημα _máthēma_, “knowledge, study, learning”) is the study of topics such as quantity (numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . Mathematicians seek out patterns and use them to formulate new conjectures . Mathematicians resolve the truth or falsity of conjectures by mathematical proof . When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic , mathematics developed from counting , calculation , measurement , and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry
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Harold Scott MacDonald Coxeter
HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London but spent most of his adult life in Canada . He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the _Canadian Music Journal_. He worked for 60 years at the University of Toronto and published twelve books . He was most noted for his work on regular polytopes and higher-dimensional geometries
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Group (mathematics)
In mathematics , a GROUP is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms , namely closure , associativity , identity and invertibility . One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry
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Group Presentation
In mathematics , one method of defining a group is by a PRESENTATION. One specifies a set S of GENERATORS so that every element of the group can be written as a product of powers of some of these generators, and a set R of RELATIONS among those generators. We then say G has presentation S R . {displaystyle langle Smid Rrangle .} Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation a a n = 1 . {displaystyle langle amid a^{n}=1rangle .} where 1 is the group identity
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Kaleidoscope
A KALEIDOSCOPE is an optical instrument with two or more reflecting surfaces inclined to each other in an angle, so that one or more (parts of) objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection . The reflectors (or mirrors ) are usually enclosed in a tube, often containing on one end a cell with loose, colored pieces of glass or other transparent (and/or opaque) materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing viewed pattern
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Reflection Group
In group theory and geometry , a REFLECTION GROUP is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space . The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups . While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem ), it is a continuous group (indeed, Lie group ), not a discrete group, and is generally considered separately. CONTENTS * 1 Definition * 2 Examples * 2.1 Plane * 2.2 Space * 3 Kaleidoscopes * 4 Relation with Coxeter groups * 5 Finite fields * 6 Generalizations * 7 See also * 8 References * 9 External links DEFINITIONLet E be a finite-dimensional Euclidean space
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Symmetry Group
In group theory , the SYMMETRY GROUP of an object (image , signal , etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric , it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry , but the concept may also be studied in more general contexts as expanded below. CONTENTS * 1 Introduction * 2 One dimension * 3 Two dimensions * 4 Three dimensions * 5 Symmetry groups in general * 6 See also * 7 Further reading * 8 External links INTRODUCTIONThe "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern . The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors
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Regular Polyhedron
A REGULAR POLYHEDRON is a polyhedron whose symmetry group acts transitively on its flags . A regular polyhedron is highly symmetrical, being all of edge-transitive , vertex-transitive and face-transitive . In classical contexts, many different equivalent definitions are used; a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex . A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra, known as the Platonic solids . These are the: tetrahedron {3, 3}, cube {4, 3}, octahedron {3, 4}, dodecahedron {5, 3} and icosahedron {3, 5}. There are also four regular star polyhedra , making nine regular polyhedra in all
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Regular Polytope
In mathematics , a REGULAR POLYTOPE is a polytope whose symmetry group acts transitively on its flags , thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube ). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures . These two conditions are sufficient to ensure that all faces are alike and all vertices are alike
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Weyl Group
In mathematics , in particular the theory of Lie algebras , the WEYL GROUP of a root system Φ is a subgroup of the isometry group of the root system . Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group . Abstractly, Weyl groups are finite Coxeter groups , and are important examples of these. The Weyl group
Weyl group
of a semi-simple Lie group , a semi-simple Lie algebra , a semi-simple linear algebraic group , etc. is the Weyl group
Weyl group
of the root system of that group or algebra . It is named after Hermann Weyl
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Simple Lie Algebra
In group theory , a SIMPLE LIE GROUP is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups . A SIMPLE LIE ALGEBRA is a non-abelian Lie algebra whose only ideals are 0 and itself (or equivalently, a Lie algebra of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups
Lie groups
are a class of Lie groups
Lie groups
which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups
Lie groups
are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative
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Triangle Group
In mathematics , a TRIANGLE GROUP is a group that can be realized geometrically by sequences of reflections across the sides of a triangle . The triangle can be an ordinary Euclidean triangle, a triangle on the sphere , or a hyperbolic triangle . Each triangle group is the symmetry group of a tiling of the Euclidean plane , the sphere , or the hyperbolic plane by congruent triangles, a fundamental domain for the action, called a Möbius triangle . CONTENTS * 1 Definition * 2 Classification * 2.1 The Euclidean case * 2.2 The spherical case * 2.3 The hyperbolic case * 2.3.1 Hyperbolic plane * 3 Von Dyck groups * 4 Overlapping tilings * 5 History * 6 Applications * 7 See also * 8 References * 9 External links DEFINITIONLet l, m, n be integers greater than or equal to 2
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Euclidean Plane
In physics and mathematics , TWO-DIMENSIONAL SPACE or BI-DIMENSIONAL SPACE is a geometric model of the planar projection of the physical universe . The two dimensions are commonly called length and width. Both directions lie in the same plane . A sequence of _n_ real numbers can be understood as a location in _n_-dimensional space. When _n_ = 2, the set of all such locations is called two-dimensional space or bi-dimensional space, and usually is thought of as a Euclidean space
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Hyperbolic Space
In mathematics , HYPERBOLIC SPACE is a homogeneous space that has a constant negative curvature , where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions , and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry , and elliptic geometry that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point . Another distinctive property is the amount of space covered by the n-ball in hyperbolic n-space: it increases exponentially with respect to the radius of the ball for large radii, rather than polynomially
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Kac–Moody Algebra
In mathematics , a KAC–MOODY ALGEBRA (named for Victor Kac and Robert Moody , who independently discovered them) is a Lie algebra , usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix
Cartan matrix
. These algebras form a generalization of finite-dimensional semisimple Lie algebras , and many properties related to the structure of a Lie algebra such as its root system , irreducible representations , and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of KAC–MOODY ALGEBRAS called AFFINE LIE ALGEBRAS is of particular importance in mathematics and theoretical physics , especially two-dimensional conformal field theory and the theory of exactly solvable models
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