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Topic: Symmetry group

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Infinite cyclic group

Frieze group

Crystallographic point group

Crystallographic group

Space group

3D crystallographic group

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Group (mathematics)

Elementary group theory

Group action

Lie group

Symmetric group

Symmetry

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	Symmetry group - Wikipedia, the free encyclopedia
		Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point.
		The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.
		This is the symmetry group of a circle.
	en.wikipedia.org /wiki/Symmetry_group (1546 words)

	Symmetry - Wikipedia, the free encyclopedia
		For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space.
		Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, and ornamentation; many facades adhere to bilateral symmetry.
		Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord.
	en.wikipedia.org /wiki/Symmetry (2564 words)

	AllRefer.com - symmetry (Physics) - Encyclopedia
		symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences.
		The symmetry groups of three-dimensional figures are of special interest because of their application in fields such as crystallography (see crystal).
		In biology, symmetry is studied in the correspondences between different parts of a given organism, as between the left and right halves of the human body or between the various segments of a starfish (see symmetry, biological).
	reference.allrefer.com /encyclopedia/S/symmetry.html (428 words)

	Learn more about Symmetry in the online encyclopedia. (Site not responding. Last check: 2007-10-22)
		The most familiar and conventionally taught type of symmetry is the left-right or mirror image symmetry exhibited for instance by the letter T: when this letter is reflected along a vertical axis, it appears the same.
		For a geometrical object, this is known as its symmetry group; for an algebraic object, one uses the term automorphism group.
		The generalisation of symmetry in physics to mean invariance under any kind of transformation has become one of the most powerful tools of theoretical physics.
	www.onlineencyclopedia.org /s/sy/symmetry.html (1201 words)

	Wallpaper Groups: the 17 plane symmetry groups
		The lattice is rectangular, and a quarter-rectangle of a fundamental region for the translation group is a fundamental region for the symmetry group.
		The lattice is rhomic, and a quarter of a fundamental region for the translation group is a fundamental region for the symmetry group.
		The lattice is square, and an eighth, a triangle, of a fundamental region for the translation group is a fundamental region for the symmetry group.
	www.clarku.edu /~djoyce/wallpaper/seventeen.html (970 words)

	Symmetry and Symmetry Breaking
		The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in the concept of symmetry in physics that took place in the twentieth century.
		According to Curie, symmetry breaking has the following role: for the occurrence of a phenomenon in a medium, the original symmetry group of the medium must be lowered (broken, in today's terminology) to the symmetry group of the phenomenon (or to a subgroup of the phenomenon's symmetry group) by the action of some cause.
		Symmetries may be used to explain (i) the form of the laws, and (ii) the occurrence (or non-occurrence) of certain events (this latter in a manner analogous to the way in which the laws explain why certain events occur and not others).
	plato.stanford.edu /entries/symmetry-breaking (9818 words)

	Symmetry Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-22)
		The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the geometric kites and the isosceles trapezoids.
		Group cm can also be described as a rectangular checkerboard pattern, where the pattern of each of the two tiles is symmetric in, say, the horizontal direction, or looking at it differently (by shifting half a tile) a checkerboard pattern where the two tiles are each other's mirror image.
		Group cmm can be described as a checkerboard pattern of 2-fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile in both directions) a checkerboard pattern of two horizontally and vertically symmetric tiles.
	encyclopedia.localcolorart.com /encyclopedia/Symmetry (4068 words)

	Groups and Symmetry
		In contrast, group theorists, such as Webb, use group actions on a set to study the structure of the group.
		Often, the actions of the group are symmetries of the object.
		The above example demonstrates that by studying the subgroups of a group which fix given parts of a simplicial complex and by considering the geometry of that complex, we are able to gain information about the structure of the group.
	www.geom.uiuc.edu /docs/forum/groups_symmetry (907 words)

	Symmetry and Patterns Page (Site not responding. Last check: 2007-10-22)
		Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions.
		The theory is applied to an experiment consisting of a system of four coupled oscillators with the symmetry of the permutation group on four symbols.
		In the analysis of the linear stability of basic states in fluid mechanics that are slowly varying in space, the quasi-homogeneous hypothesis is often invoked, where the stability exponents are defined locally and treated as slowly varying functions of a spatial coordinate.
	www.maths.surrey.ac.uk /research/SYMMETRY (2149 words)

	The Geometry Junkyard: Symmetry and Group Theory
		Associating the symmetry of the Platonic solids with polymorf manipulatives.
		Ron Lifshitz provides a light introduction to the symmetry of periodic and aperiodic crystals, and the complications introduced by including permutations of colors in a coloring as part of a symmetry operation.
		Group theoretic mathematics for determining whether a polygon formed out of hexagons can be dissected into three-hexagon triangles, or whether a polygon formed out of squares can be dissected into restricted-orientation triominoes.
	www.ics.uci.edu /~eppstein/junkyard/sym.html (1155 words)

	Group Theory
		Each of these Symmetry Operations is associated with a Symmetry Element which is a point, a line, or a plane about which the operation is performed such that the molecule's orientation and position before and after the operation are indistinguishable.
		The possible Symmetry Operations associated with a molecule are determined by the Symmetry Elements possessed by that molecule.
		Therefore the first step in applying Group Theory to molecular properties is to identify the complete set of Symmetry Elements possessed by the molecule.
	www.science.siu.edu /chemistry/tyrrell/group_theory/sym1.html (487 words)

	Crystallographic point group (Site not responding. Last check: 2007-10-22)
		In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged.
		The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties.
		The point groups are denoted by their component symmetries.
	www.sciencedaily.com /encyclopedia/crystallographic_point_group (342 words)

	The Sherrill Group: Research: Symmetry Breaking
		The first thing one should examine is the symmetry of the wavefunction at the high-symmetry geometry: if the wavefunction fails to exhibit the full symmetry of the point group, this is a sign that predictions of geometrical symmetry breaking may be artifactual.
		Paradoxically, for a given approximation scheme, such symmetry broken wavefunctions can be lower in energy (and hence better in a variational sense) than symmetric wavefunctions, even though the exact wavefunction must be symmetric.
		Many users of quantum chemistry programs probably assume that all such results are due to bugs in the code (or failing to follow the correct solution for one of the geometries, which frequently happens), but this is not always true: they may result from fundamental failures in the underlying quantum mechanical models.
	vergil.chemistry.gatech.edu /research/symbrk/symbrk.html (615 words)

	Symmetry and Point Groups (Site not responding. Last check: 2007-10-22)
		Symmetry elements are geometric entities that are used to minipulate molecules so as to transform them from one spatial orientation into another, indistinguishable, orientation.
		The point group or symmetry group is the name given to the collection of symmetry elements possessed by a molecule.
		It also is the symmetry (approximately) possesed by most of us critters who go about on the surface of the earth.
	chemistry.umeche.maine.edu /Modeling/symmetry.html (1161 words)

	Symmetry Group
		Thus when we are considering the symmetry group of such a figure we may suppose G to be generated by rotations and, perhaps, reflexions.
		It is easy to see why the symmetry groups are the same; for the centers of the faces of a cube are the vertices of a regular octahedron, and the centers of the faces of a regular octahedron are the vertices of a cube.
		It is a matter of great interest and relevance here that the symmetries of the Diagonal Cube and the special braided octahedron of Figure 7 and Figure 16, respectively (of [Rec]) each permute the four braided strips from which the models are made.
	www.mi.sanu.ac.yu /vismath/hil/ped3.htm (1390 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		The definition of the symmetry group as a group of unitary operators that commute with the hamiltonian is insufficient.
		For this purpose it is sufficient to choose a group with the required dimensions of irreducible representations.
		The symmetry group of an anisotropic oscillator is determined.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4689578 (164 words)

	PlanetMath: bifurcation problem with symmetry group
		be a Lie group acting on a vector space
		"bifurcation problem with symmetry group" is owned by Daume.
		This is version 1 of bifurcation problem with symmetry group, born on 2003-08-21.
	planetmath.org /encyclopedia/BifurcationProblemWithSymmetryGroup.html (95 words)

	Point Group Symmetry
		Introduction and notes to symmetry operations and dipole transitions.
		Categorisation of point groups crystal class (cubic, tetragonal etc.).
		Some references for symmetry in physics and chemistry.
	www.phys.ncl.ac.uk /staff/njpg/symmetry (60 words)

	3.2 The Symmetry Group of Friezes
		is the symmetry group of friezes (abbreviated: the frieze group) if it's translational subgroup is generated by one translation.
		By choosing possible orthogonal groups for the frieze groups, respecting the condition that it must leave the lattice invariant (Theorem 3.1) we will demonstrate that there are 7 non-isomorphic frieze groups.
		For completing the classification of frieze groups, we need to show that among the 7 groups derived there are no mutually isomorphic ones.
	www.mi.sanu.ac.yu /vismath/ana/ana5.htm (464 words)

	QUANTUM GROUP SYMMETRY AND q-TENSOR ALGEBRAS
		Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics.
		This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator.
		Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction.
	www.worldscibooks.com /physics/2815.html (205 words)

	JCE 2000 (77) 313 [Mar] Molecular Symmetry and Group Theory (by Robert L. Carter) (Site not responding. Last check: 2007-10-22)
		The introduction to group theory in the first four chapters neatly interweaves basic arguments needed to solve chemical problems in inorganic chemistry.
		While the focus of the text is to present group theory without a full mathematical derivation, connections to the mathematical underpinnings are suggested.
		One of the themes in Molecular Symmetry and Group Theory is group-subgroup relationships.
	jchemed.chem.wisc.edu /Journal/Issues/2000/mar/abs313_1.html (701 words)

	Introduction to Symmetry (Science U)
		There are actually four distinct kinds of symmetry, corresponding to four basic ways of moving a tile around in the plane, illustrated to the right.
		This is simply because if we do one symmetry followed by another, then we could have just move the tiling directly from its initial postion to its final position, and it would still match up.
		The symmetry group of a tiling is just the collection of all its symmetries.
	www.scienceu.com /geometry/articles/tiling/symmetry.html (812 words)

	Knot Table: Symmetry Group (Site not responding. Last check: 2007-10-22)
		If a knot is viewed as a pair, (S^3,K), its symmetry group is defined to be the group of diffeomorphisms of the pair, modulo the normal subgroup generated by those diffeomorphisms that are isotopic to the identity.
		For a hyperbolic knot the symmetry group is always finite, given by the group of isometries of the unique complete hyperbolic structure on the complement.
		For torus knots the symmetry group is always of order 2.
	www.indiana.edu /~knotinfo/descriptions/full_symmetry_group.html (142 words)

	Symmetry Group (Site not responding. Last check: 2007-10-22)
		Symmetry theory coincides with the automorphism-transformation group theory: the group of transformation preserving the structural integrity of the systems under consideration (Weyl 1952).
		Since the composition of any two symmetry transformation is another symmetry transformation, the set of all such symmetry transformations is a group which is called the group of symmetry of S relative to the partition P.
		are two symmetry transformation of S relative to the partition P because all the arrows start and end in the same class of the partition, while the transformation t
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node167.html (157 words)

	Beliefs-Preferences Gauge Symmetry Group and Replication of Contingent Claims in a General Market Environment
		Beliefs-Preferences Gauge Symmetry Group and Replication of Contingent Claims in a General Market Environment studies the actual financial phenomena underlying the evaluation of financial derivatives, which is today virtually identified with and even replaced by the study of the mathematical aspects of stochastic calculus as a model for such phenomena.
		This volume introduces a fundamental symmetry, a gauge symmetry, between beliefs of market participants and their preferences in a general market environment for a market with exchange of an arbitrary number of arbitrary underlying securities.
		In particular, this beliefs-preferences gauge symmetry makes it possible to obtain an evolution equation that determines, in a general market environment, the values of European contingent claims independent of these beliefs and preferences.
	www.ieslc.com /bpgsg.html (559 words)

	Amazon.com: Books: Symmetry (Site not responding. Last check: 2007-10-22)
		Theory of Groups and Quantum Mechanics by H.
		Weyl presents a masterful and fascinating survey of the applications of the principle of symmetry in sculpture, painting, architecture, ornament, and design; its manifestations in organic and inorganic nature; and its philosophical and mathematical significance.
		Of course much of what he says about symmetry is true of aesthetics and beauty in general and many parallels can be drawn between what he is saying and other items like music that may not appear to have clear symmetry right off the bat.
	www.amazon.com /exec/obidos/tg/detail/-/0691023743?v=glance (934 words)

	symmetries
		Then someone shook up the world by pointing out that it has as symmetries the group E(3), since up and down are in fact merely conventional concepts and one man's up is another woman's down.
		The group consisting of the Poincaré group and dilations is sometimes called the "Weyl group".
		unlike the previous groups, the diffeomorphism group is often regarded as a "gauge" group, that is, a group expressing the fact that the mathematics you have used to express the physics contains redundancies - i.e., two spacetimes with metrics that are equivalent under some diffeomorphism should really be regarded as the same physical system.
	math.ucr.edu /home/baez/symmetries.html (2452 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Identifying which combinations of these elements are present in a molecule will enable you to classify the group symmetry.
		A symmetry element is some feature of the molecule relative to which an operation can be performed, like rotation around an axis.
		The "point-group" symmetry is a method of denoting the combination of symmetry elements which a molecule contains.
	www.shodor.org /succeed/compchem/symtry.html (368 words)

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