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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 ). 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 Kac–Moody algebras . Standard references include (Humphreys 1992 ) and (Davis 2007 )
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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. Rigorous arguments first appeared in Greek mathematics , most notably in Euclid 's _Elements _. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century , it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions
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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. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College , Cambridge in 1926 to read mathematics. There he earned his BA (as Senior Wrangler ) in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow , where he worked with Hermann Weyl , Oswald Veblen , and Solomon Lefschetz
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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 . For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other
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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. This may be written equivalently as a a n , {displaystyle langle amid a^{n}rangle ,} since terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called RELATORS, distinguishing them from the relations that include an equals sign. Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group. A closely related but different concept is that of an absolute presentation of a group
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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. CONTENTS * 1 Etymology * 2 Principles * 3 History * 4 Variations * 4.1 General variables * 4.2 Different versions suggested by Brewster * 4.3 Later variations * 5 Publications * 6 Applications * 7 See also * 8 References * 9 External links ETYMOLOGYCoined in 1817 by Scottish inventor David Brewster , "kaleidoscope" is derived from the Ancient Greek καλός (kalos), "beautiful, beauty", εἶδος (eidos), "that which is seen: form, shape" and σκοπέω (skopeō), "to look to, to examine", hence "observation of beautiful forms." PRINCIPLES The basic configuration of reflecting surfaces in the kaleidoscope, as illustrated in the 1817 patent. Fig. 2 and Fig. 3 show alternative shapes of the reflectors
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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 . A FINITE REFLECTION GROUP is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An AFFINE REFLECTION GROUP is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields , leading to COMPLEX REFLECTION GROUPS and analogues of reflection groups over a finite field
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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
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. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a group action on objects in it. The symmetry group is sometimes also called FULL SYMMETRY GROUP in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations ) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e
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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. Note, however, that this definition does not work for abstract polytopes . A regular polytope can be represented by a Schläfli symbol
Schläfli symbol
of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}
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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
Hermann Weyl
. CONTENTS * 1 Weyl chambers * 2 Coxeter group structure * 2.1 Example * 3 Weyl groups in algebraic, group-theoretic, and geometric settings * 4 Bruhat decomposition * 5 Analogy with algebraic groups * 6 Cohomology * 7 See also * 8 Footnotes * 8.1 Notes * 8.2 Citations * 9 References * 10 Further reading * 11 External links WEYL CHAMBERSRemoving the hyperplanes defined by orthogonality to the roots of Φ cuts up Euclidean space
Euclidean space
into a finite number of open regions, called WEYL CHAMBERS. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive . In particular, the number of Weyl chambers equals the order of the Weyl group
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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
Lie algebra
whose only ideals are 0 and itself (or equivalently, a Lie algebra
Lie algebra
of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups are a class of 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 are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative . Together with the commutative Lie group of the real numbers, R {displaystyle mathbb {R} } , and that of the unit complex numbers, U(1), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension . Many commonly encountered Lie groups are either simple or close to being simple: for example, the group SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1. An equivalent definition of a simple Lie group follows from the Lie correspondence : a connected Lie group is simple if its Lie algebra
Lie algebra
is simple
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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. A TRIANGLE GROUP Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians )
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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 . CONTENTS * 1 History * 2 In geometry * 2.1 Coordinate systems * 2.2 Polytopes * 2.2.1 Convex * 2.2.2 Degenerate (spherical) * 2.2.3 Non-convex * 2.3 Circle
Circle
* 2.4 Other shapes * 3 In linear algebra * 3.1 Dot product, angle, and length * 4 In calculus * 4.1 Gradient
Gradient
* 4.2 Line integrals and double integrals * 4.3 Fundamental theorem of line integrals * 4.4 Green\'s theorem * 5 In topology * 6 In graph theory * 7 References * 8 See also HISTORYBooks I through IV and VI of Euclid\'s Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas , parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics
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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
Euclidean geometry
, and elliptic geometry that have a constant positive curvature. When embedded to a Euclidean space
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. CONTENTS * 1 Formal definition * 2 Models of hyperbolic space * 2.1 The hyperboloid model * 2.2 The Klein model
Klein model
* 2.3 The Poincaré ball model * 2.4 The Poincaré half space model * 3 Hyperbolic manifolds * 3.1 Riemann surfaces * 4 See also * 5 References FORMAL DEFINITIONHYPERBOLIC N-SPACE, denoted Hn, is the maximally symmetric, simply connected , n-dimensional Riemannian manifold with a constant negative sectional curvature . Hyperbolic space
Hyperbolic space
is a space exhibiting hyperbolic geometry . It is the negative-curvature analogue of the n-sphere
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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 . 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 . Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities , which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated that Rogers–Ramanujan identities can be derived in a similar fashion
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