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Topic: Crystallographic group

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Symmetry group

3D crystallographic group

Group theory

Group (mathematics)

Crystal structure

Symmetry

Space group

Crystallographic point group

	Crystallographic Topology - Appendix A
		There are 1651 groups in the complete bicolor space group family, 230 of which are the regular space groups with fl symmetry operators, another 230 have simultaneously fl and white (gray) symmetry operators, and the remaining 1191 nontrivial bicolor space groups have mixed fl and white operators.
		Bicolor groups are called magnetic groups when used to describe simultaneously the arrangement of atoms (regular symmetry) and the up/down magnetic spin vector orientations (antisymmetry) for magnetic atoms in a crystal (Opechowski and Guccione, 1965).
		Since the group generators for space groups are given explicitly in the crystallographic space group nomenclature, the bicolor space group nomenclature is given by simply placing flags, such as , on the antisymmetry operators in the group* name (see Table A.1).
	www.ornl.gov /sci/ortep/topology/app_a.html (1795 words)

	[No title]
		The groups p3m1 and p31m have two lines connecting them since p3m1 is a normal subgroup of index 3 in p31m while p31m is a subgroup of index 3 in p3m1 but it is not a normal subgroup.
		In Lattice 2, all of the crystallographic groups, which lie in p4m along with their minimal indices, are shown.
		Subgroup Lattice 7 categorizes the crystallographic groups in terms of rotations, reflections and glide reflections.
	members.tripod.com /vismath8/tennant/index.html (1143 words)

	[cryst] 2 Affine crystallographic groups
		For instance, if the lower Wyckoff position consists of a space group orbit of lines (and thus the upper one of an orbit of points), the label of the connection line is the number of lines in the orbit which cross a fixed representative point of the upper Wyckoff position.
		in the group of all unimodular transformations of the lattice spanned by the
		Space groups with a centered lattice are therefore given in the non-primitive basis crystallographers are used to.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/cryst/htm/CHAP002.htm (2609 words)

	Description of a Magnetic Structure
		In CCSL magnetic structures are described using a propagation vector to define the periodicity, and a magnetic space group to define the relative orientations of spins on different sub-lattices within the non-magnetic unit cell.
		The magnetic space group must be congruent with the crystallographic space group or one of its sub-groups.
		Each of the elements of the magnetic group acts on the magnetic moment with the rotation and translation appropriate to the corresponding element in the crystallographic group.
	www.ill.fr /dif/ccsl/mk4man/c6node1.html (243 words)

	[No title]
		Let me just remark that our group theoretic program system GAP can be of help in such considerations since on one hand it offers possibilities to calculate with such factor groups and on the other hand with its last release has tables of all space groups (not only in three but also in four dimensions).
		A 3-D crystallographic group is a group acting on 3-dimensional Euclidean space (ordinary space) in such a way that the quotient space is compact, or equivalently, that there is a bounded fundamental domain for the action.
		These groups (as well >>crystallographic groups in any dimension, >>as all "genuine" 3-orbifold groups) are residually finite, meaning that >>they have many finite quotient groups, almost certainly enough quotients >>to establish the correct minimum number of generators.
	www.math.niu.edu /~rusin/known-math/94/crystallo (2132 words)

	Machines for building symmetry [1]
		groups in one dimension: there is only one finite group, corresponding to just one mirror; there is also only one infinite group, corresponding to two parallel mirrors.
		So, the wallpaper groups which can be seen in a mirror box are seven, the four ones which are generated by reflections, and other three, which contain a subgroup generated by reflections.
		Instead, in the caleidoscope associated to the symmetry group G of the tetrahedron we may see either objects having G as symmetry group or objects having the same symmetry group H of a cube (as G is a subgroup of index 2 in H).
	arpam.free.fr /dedo.htm (4459 words)

	Service Crystallography
		The purpose of this group of the American Crystallographic Association (hereafter ACA) is to further the advancement of all aspects of the application of service crystallographic techniques and to promote communication between persons interested in such techniques and results.
		The activities of this Group shall be administered by an Executive Committee of three officers elected by members of the Group.
		The Chair-Elect shall also serve to represent the Group to the Council of the Association and to be the Group's representative on the Program Committee of ACA meetings and to perform such duties and functions on behalf of the Group as the Chairperson, may, from time to time, direct.
	aca.hwi.buffalo.edu /ServiceCrystallography.html (635 words)

	[aclib] 1 The Almost Crystallographic Groups Package
		The output is a pcp group isomorphic to the corresponding matrix group.
		For the 3-dimensional almost crystallographic groups the type is a string representing the numbers from 1 to 17, i.e.
		For the 4-dimensional almost crystallographic groups with a Fitting subgroup of class 2 the type is a string of 3 or 4 characters.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/aclib/htm/CHAP001.htm (2073 words)

	Structural Biology and Synchrotron Radiation: Evaluation of Resources and Needs (Site not responding. Last check: 2007-10-21)
		Thus, in 1997, use of synchrotron radiation for crystallographic experiments by non-specialists is limited to biologists who are able to recruit and pay for crystallographic expertise within their research programs.
		Even when the speaker was not a crystallographer, frequent references were made to protein structures determined through collaborations with a crystallographic group.
		This integration of crystallographic research into the framework of a modern molecular biology laboratory is a phenomenon of the last five years and is clearly growing rapidly, although the numbers are difficult to project with any accuracy.
	genome.rtc.riken.go.jp /hgmis/biosync/body.html (3771 words)

	Crystal Structures (Site not responding. Last check: 2007-10-21)
		The crystals in Figures 1.1a and 1.1b have equivalent symmetry groups, while some of the symmetries of the honeycomb lattice are different.
		For a complex structure the identification of the symmetry group may be a rather nontrivial task.
		Considering the examples in Figure 1.1, the rotations around P and the mirror line m are point group symmetries, but the combination of mirroring around m' and the subsequent shift is not a point group operation.
	solidstate.physics.sunysb.edu /book/prob/node3.html (2041 words)

	GAP Manual: 60 CrystGap--The Crystallographic Groups Package (Site not responding. Last check: 2007-10-21)
		Note that this is different from the crystallographic convention, where matrices usually act from the left on column vectors (see also The Crystallographic Groups Library).
		Space groups which are equivalent under conjugation in the affine group (shortly: affine equivalent space groups) are said to belong to the same space group type.
		For space groups with larger point groups, most of the time in the computation of Wyckoff positions (see WyckoffPositions) is spent computing the subgroup lattice of the point group.
	www.math.jussieu.fr /~jmichel/htm/CHAP060.htm (2847 words)

	WULFFMAN - CTCMS
		The simplest example of a group is a permutation group, the set of all permutations of a collection of objects.
		A point group is a group whose elements represent point isometries, i.e., actions on Euclidean space that leave the distance between points invariant.
		Vectors a, b, and c are referred to as the translation vectors of the lattice, and define a nontrivial, discrete subgroup of the group of all translations.
	www.ctcms.nist.gov /wulffman/docs_1.2 (4384 words)

	Tessellation Summary
		For example, the symmetry group of the standard tiling by 1x1 squares is equal to the set of all compositions of the following two maps and their inverses: the map that takes any arbitrary point (x, y) to (x + 1,y) and the map that takes (x, y) to (x, y + 1).
		The crystallographic groups are the symmetry groups for which there is a fundamental domain with finite area (or volume).
		Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two.
	www.bookrags.com /Tessellation (3766 words)

	Crystallographic Topology 101 - Overview
		The structural crystallography of interest involves the group theory required to describe symmetric arrangements of atoms in crystals, and a classification of the simplest arrangements as lattice complexes.
		The geometric topology of interest is the topological properties of crystallographic groups, represented as orbifolds, and the Morse theory global analysis of critical points in symmetric functions.
		A crystallographic orbifold, Q, may be formally defined as the quotient space of a sphere, S, or Euclidean, E, space modulo a discrete crystallographic symmetry group, G (i.e., Q=K/G where K=S or E).
	www.ornl.gov /ortep/topology/overview.html (2580 words)

	2.4 Wallpaper Groups
		A group of symmetries of the plane that is doubly infinite is a crystallographic group, or wallpaper group.
		The Conway notation for wallpaper groups is based on what types of non-translational symmetries occur in the ``simplest description'' of the group: * indicates a reflection (mirror symmetry), × a glide-reflection, and a number n indicates a rotational symmetry of order n (rotation by 360°/n).
		The last column of the table gives the number of degrees of freedom in the choice of the group, that is, the dimension of the space of inequivalent groups of the given type (equivalent groups are those that can be obtained from one another by proportional scaling or rigid motion).
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node12.html (596 words)

	Table of contents for Library of Congress control number 2005050093
		Representation of the symmetric group by permutation matrices 17 6.
		Formulae giving those characters of the symmetric group on m letters which are attached to two and three element parti- tions of m in terms of the class numbers (a).
		The analysis of the representations of the real orthogonal group which are furnished, by the principle of selection, by the irreducible representations of the full linear group.
	www.loc.gov /catdir/toc/fy0604/2005050093.html (645 words)

	METHODS OF PERFECT COLORING
		All the plane periodic mosaics may be classified by means of the determination of their symmetry group, having 17 different classes as a result [1], although from an artistic point of view their number may be infinite.
		However, if we bear in mind the colors, the number of groups which determine the symmetry of the mosaic is greater [10].
		by the group of the permutations of G/G
	members.tripod.com /vismath2/ruiz2/index.html (1720 words)

	Finite and Affine Coxeter Groups
		An affine reflection group is a group generated by reflections in affine space (in other words, real reflections in a hyperplane that does not necessarily pass through the origin).
		A Coxeter group is called affine if it is infinite and it has a representation as a discrete, properly acting, affine reflection group (see [Bou68] for more details on discreteness and proper action).
		A Coxeter group is finite if, and only if, all its irreducible components are finite; a Coxeter group is affine if, and only if all its irreducible components are finite or affine, and at least one component is affine.
	www.math.lsu.edu /magma/text982.htm (1101 words)

	The Structural Condensed Matter Physics Group (Site not responding. Last check: 2007-10-21)
		The Physical Crystallography Group was formed in 1943 as the X-ray Analysis Group of The Institute of Physics and is now a joint group of The Institute of Physics and the British Crystallographic Association.
		The Group was renamed the Structural Condensed Matter Physics Group in 1999, to reflect the wider scope of its subject area within the physics community.
		The Physical Crystallography Group played a major role in the formation in 1982 of the British Crystallographic Association which provides a focus for the exchange of ideas between people from a wide range of disciplines and represents the interests of the crystallographic community nationally.
	www.iop.org /Our_Activities/Groups_and_Divisions/Subject_Groups/Structural_Condensed_Matter_Physics/page_3541.html (279 words)

	CHROMATIC PLANE COMPOSITIONS (Site not responding. Last check: 2007-10-21)
		The main theorem of the article is the Theorem 2 which by means of the color equation characterizes the successions of subgroups of a crystallographic group which may be the isotropy color groups of a chromatic plane periodic composition.
		The regularity is described when we establish that the isotropy groups of the two different colors have to be comparable; in such a way that they form a succession of subgroups of the group used for the construction of the discolored mosaic.
		The main theorem of the article is the theorem which by means of the color equation characterizes the successions of subgroups of a crystallographic group which may be the isotropy color groups of a chromatic plane periodic composition.
	www.mi.sanu.ac.yu /vismath/ruiz1/index.html (3705 words)

	Crystallographic site-symmetry operations
		In this manuscript the term site-symmetry group or, for short, site symmetry, is preferred for reasons which will become clear in Section 3.4.
		For the description of the crystallographic symmetry operations, it is convenient to have available the notion of the `order of an isometry'.
		Due to the periodicity of the crystals, the rotation angles of crystallographic rotations are restricted to multiples of
	www.iucr.org /iucr-top/comm/cteach/pamphlets/22/node20.html (558 words)

	JLT 1404 (Site not responding. Last check: 2007-10-21)
		An essential crystallographic set of isometries can be endowed with a crystallographic pseudogroup structure.
		Under certain well chosen conditions on the essential crystallographic set of isometries Γ we show that the elements in Γ define a crystallographic group G, and an embedding Φ from Γ to G exists which is an almost isomorphism close to the identity map.
		This can be interpreted as a sort of metric rigidity of crystallographic groups: if there is an essential crystallographic set of isometries which is metrically close to an inner part of a crystallographic group, then there exists a local homomorphism-preserving embedding in this crystallographic group.
	www.heldermann.de /JLT/JLT14/jlt1404.htm (209 words)

	Talk at WASGAS --- August 1995
		If the realization of the Coxeter group is crystallographic, which means that it preserves a lattice, then it is the Weyl group of a Kac-Moody group, and the structure of the graph of positive roots should have something to do with the structure of the group.
		To every Coxeter group is associated a Hecke algebra over a polynomial ring in one variable, with generators the same as the group and similar relations.
		The significance of cells is not generally understood, but (again according to Lusztig) if the Coxeter group is affine, the two-sided cells correspond to the semi-regular unipotent class in the associated semi-simple complex group and the left cells parametrize homology classes of the subvariety of Borel subgroups containing a fixed element of that class.
	www.math.ubc.ca /~cass/research/wasgas-talk.html (821 words)

	SC680 - Crystallographic Data (Site not responding. Last check: 2007-10-21)
		The crystallographic coordinates are held in the Brookhaven Protein Data Bank (PDB) and are available to researchers via computer via gopher (gopher://pdb.pdb.bnl.gov) or via the web (http://www.pdb.bnl.gov).
		When studying how unit cells are packed together it is necessary to know the crystallographic space group in order to be able to work out the position of the adjacent molecules in adjacent unit cells and build up a lattice of molecules if you so choose.
		If using crystallographic information as a starting point for molecular modelling it is sensible to assess the quality of the crystal structure before use.
	www.chem.swin.edu.au /courses/swin/sc680/crystal1.html (1476 words)

	Combinatorial Tiling Theory
		a discrete co-compact group of isometries of the euclidean plane, the sphere or the hyperbolic plane, enumerate all 2-dimensional equivariant tilings (T,G) that are tile-N-transitive, i.e.
		In "Two-Dimensional Groups, Orbifolds and Tilings" (Balke and Huson 1996) we formalize this and prove that the symmetries of the orbifold symbol (Macbeath 1967, Conway 1990) of a 2-dimensional symmetry group determine how many tilings exist for that group.
		The 219 isomorphism types of space groups give rise to precisely 189 different orbifold graphs, only 14 of which correspond to more than one group, as described in "Orbifold Triangulations and Crystallographic Groups" (Delgado and Huson 1997).
	www-ab.informatik.uni-tuebingen.de /people/delgado/Old-Stuff/Tiling-Theory/index.html (3565 words)

	Structure of Periodic and Aperiodic Crystals
		This is a home page of the research group of Dr. Vaclav Petricek in the Institute of Physics.
		New page of the Czech and Slovak Crystallographic Association is now maintained at www.xray.cz.
		This laboratory is specialized in structure analysis of aperiodic and periodic crystals.
	www-xray.fzu.cz /xraygroup/www/xraygroup.html (277 words)

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