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Topic: Translational symmetry

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Symmetry group

spontaneous symmetry breaking

symmetry of second derivatives

bilateral symmetry

CP symmetry

radial symmetry

P symmetry

T symmetry

CPT symmetry

mirror symmetry

rotational symmetry

gauge symmetries

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	Translational symmetry - Wikipedia, the free encyclopedia
		Translational symmetry of an object means that a particular translation does not change the object.
		Different bases of translation vectors generate the same lattice iff one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1.
		The objects with at least the corresponding translational symmetry are the fixed points of the latter, not to be confused with fixed points of the translation of space, which are non-existent.
	en.wikipedia.org /wiki/Translational_symmetry (880 words)

	Symmetry - Wikipedia, the free encyclopedia
		Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it.
		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 (2806 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.
		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).
		As we have mentioned, there is a close connection between the notions of symmetry and equivalence, and this leads also to a notion of irrelevance: the equivalence of space points (translational symmetry) is, for example, understood in the sense of the irrelevance of an absolute position to the physical description.
	plato.stanford.edu /entries/symmetry-breaking (9818 words)

	Intro to Minerals: Crystal Class and System
		Translational symmetry describes the periodic repetition of a motif across a length or through an area or volume.
		For example, crystals of the holomorphic class of the isometric system possess inversion symmetry, three 4-fold axes of rotational symmetry, the characteristic set of four 3-fold axes of rotational symmetry which is indicative of the isometric crystal system, six 2-fold axes of rotational symmetry, and nine different mirror planes.
		It is thus apparent that the characteristic symmetry element of the isometric crystal system is the possession of four 3-fold axes of rotational symmetry, while the characteristic symmetry element of the rhombohedral division of the hexagonal crystal system is the possession of a single 3-fold axis of rotational symmetry.
	dave.ucsc.edu /myrtreia/crystal.html (4278 words)

	Symmetry (Site not responding. Last check: 2007-10-21)
		In chemistry, symmetry is used to help determine optical activity of a molecule, dipole moments, in ultraviolet and infrared spectroscopy, to determine molecular bonding, and to determine crystal structure.
		Translational symmetry is used in crystalography, and not used in molecular bonding.
		A molecule has inversion symmetry if a line can be drawn from one atom, through the center of the molecule, and come into contact with another atom that is the same distance from the center of symmetry.
	www.uvm.edu /~swgordon/131-00/131-webproj/jtremblay/symmetry.html (1237 words)

	Symmetry in Architecture by Kim Williams, Architect
		Translation of elements in one direction is found in solemn rows of soldier-like columns, or in the springing succession of arches in an aqueduct.
		Translation may also involve the repetition of entire pieces of buildings, especially in our own century, and may be one reason by modern architecture is so often referred to as boring or monotonous.
		Translational symmetry seems to carry with it an emphasis on a superlative quality in architecture: the longest, the broadest, the tallest.
	members.tripod.com /vismath/kim (4294 words)

	Symmetry as a Compositional Determinant: Chapter 2: Definitions
		Symmetry is here to be defined as a congruence which results from the operations of reflection, rotation, or translation.
		Another possible symmetry operation is a rotation around a point, or a line in three variables, exhibited by the letter Z. This letter may be rotated 180 degrees around a point to create an image identical to the original.
		Translational symmetry is often combined with reflective symmetry in music.
	solomonsmusic.net /diss2.htm (2268 words)

	Symmetry as a Compositional Determinant: IV. Translation & Rotation
		Translation is necessary here in only the temporal dimension to yield symmetry, and the interval of translation is one quarter note.
		A C major scale, for instance, may be translated at the octave to yield a congruence of pitch class.
		Symmetry may be demonstrated by rotating the figure 180° to yield a congruence.
	solomonsmusic.net /diss4.htm (1878 words)

	[No title]
		Translation Vectors (Beginning with a single lattice point, all of the other lattice points in the array can be generated by shifting the lattice point by the translation vector.
		In 2D two translation vectors are needed (three are needed in 3D) to completely describe the position of any lattice point from the lattice point at the origin: tn = n1a + n2b where a and b are the translation vectors, and n1 and n2 are integers.
		The translation vector is defined as R = ua + vb + wc Where a, b and c are the vectors which define the unit cell, and u, v and w are integers.
	www.chemistry.ohio-state.edu /~woodward/ch754/sym_2d.doc (1408 words)

	Examples of Symmetry Conservation in Images (Site not responding. Last check: 2007-10-21)
		Animals using jet propulsion and a rocket have also a cylindrical symmetry because all the directions perpendicular to their movement are dynamically the same.
		Most fish have a bilateral symmetry because their direction of displacement breaks the environmental spherical symmetry into a cylindrical symmetry which is broken by the direction of the lateral movement of the tail (propulsion) into a bilateral symmetry.
		As a first approximation, their images have the same type of translational symmetry if they are nearly parallel to the image plane, otherwise this translational symmetry is partially broken and there is a convergence in the image of the 3-space parallel line with a gradient in the pattern in the direction of these converging lines.
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node135.html (400 words)

	symmetry student page (Site not responding. Last check: 2007-10-21)
		An image has Reflectional Symmetry if there is at least one line which splits the image in half so that one side is the mirror image of the other.
		Reflectional symmetry is also called line symmetry or mirror symmetry because there is a line in the figure where a mirror could be placed, and the figure would look the same.
		Translational symmetry results from moving a figure a certain distance in a certain direction also called translating (moving) by a vector (length and direction).
	www.geom.uiuc.edu /~demo5337/s97a/students.html (228 words)

	Articles - Crystal system (Site not responding. Last check: 2007-10-21)
		A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group.
		A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector).
		In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.
	www.zdiamond.net /articles/Crystal_system (739 words)

	Mathematics and Writing in Action
		Reflective Symmetry: An object exhibits reflective symmetry when a line or axis of reflection can be drawn in the plane and the portion of the object on one side of the axis is a mirror image of the portion on the other side.
		Translational Symmetry: An object exhibits translational symmetry when a line or axis of translation can be drawn in the plane such that when one portion of the object is moved alone that line a certain number of times the entire object is obtained.
		A object exhibits glide reflective symmetry when an axis of translation and an axis of reflection, parallel to the axis of translation, can be drawn such that when one portion of the object is translated along the first axis and reflected over the second axis a certain number of times, the entire object is obtained.
	www.academic.marist.edu /mwa/frm.htm (356 words)

	LiveScience.com - Symmetry in Nature: Fundamental Fact or Human Bias?
		Symmetry is also prevalent in the physical sciences and is woven into the very laws that govern our universe.
		Translational symmetry: The laws of physics are the same whether they are acting in our solar system or at the far end of the universe.
		Symmetry is so integral to the way the universe works that Albert Einstein used it as a guiding principle when he devised his General Theory of Relativity.
	www.livescience.com /othernews/051221_symmetry_nature.html (1387 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.
		It is important to emphasize that the symmetries of the Bravais lattice are intimately related to the symmetries of the original lattice.
		The WS cell is a primitive unit cell that preserves the symmetries of the Bravais lattice.
	solidstate.physics.sunysb.edu /book/prob/node3.html (2041 words)

	Symmetry in Tessellations (Site not responding. Last check: 2007-10-21)
		A tessellation possesses translational symmetry if it can be translated by some vector and remain unchanged.
		A tessellation possesses glide reflection symmetry if it can be translated by some vector and then reflected about that vector and remain unchanged.
		A special case of glide rereflection symmetry is simple reflection or mirror symmetry, where the vector has a value of zero.
	members.cox.net /tessellations/Symmetry.html (278 words)

	Internal Structure of Crystals II
		The point symmetry of this figure then is the point symmetry of the periodic pattern of motifs.
		It is the macroscopically visible symmetry, because the translations are too small to be detected by the naked eye (We here imagine 2-D crystals to exist in reality and consisting of atoms, ions, or complexes thereof, like 3-D crystals do).
		The point symmetry of this unit mesh is 4, and it is primitive, and as such denoted by the symbol P.
	home.hetnet.nl /~heackel/d2_lattice_2.html (2775 words)

	[No title]
		In some cases however, the presence of pseudo translational symmetry or anisotropic diffraction influences the intensity statistics in such a way that twinning cannot readily be detected, even though it is present.
		The results for detection of translational symmetry via the presence of significant peaks in the native Patterson function are illustrated in Table 1.
		For the detection of pseudo translational symmetry and twinning, a similar philosophy is adopted: the summary statistics of the given data set are listed within the context of a reference set of known structures.
	www.ccp4.ac.uk /newsletters/newsletter42/articles/CCP4_2005_PHZ_RWGK_PDA.doc (2678 words)

	Lecture 5: Translational Symmetry - 2D (Site not responding. Last check: 2007-10-21)
		Many choices of translations, some are more logical or conventional.
		Symmetry Content of Planar Motifs: (Fig 3.10, O.H. 10 possible combinations: 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm (Fig 3.10) (Remember the relationships of the symbology to the mirrors and the axes.)
		Symmetry Content of Plane Lattices: (Fig 3.11, O.H. Note the symmetry which is explicit and implicit in each of the 5 plane lattices.
	www.geology.wisc.edu /~pbrown/g360/lect5.html (262 words)

	Trolling in Shallow Water (Site not responding. Last check: 2007-10-21)
		external symmetries I discussed last time were translational symmetry, which implies conservation of linear momentum; time symmetry, which implies conservation of energy; rotational symmetry, which implies conservation of angular momentum; boost symmetry, which implies conservation of a relativistic analogue of angular momentum; and time reversal and parity.
		This symmetry is called the special Lorentz symmetry, and its mathematical symbol is SL(2,c).
		These properties of symmetry operations imply that the set of operations permitted by a given symmetry form a mathematical group.
	shallows.blogspot.com /2005/02/u1xsu2xsu3-part-2.html (1856 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		--------- N.D. Mermin, D.S. Rokhsar, and D.C. Wright Beware of 46-Fold Symmetry: The Classification of Two-Dimensional Quasicrystallographic Lattices Phys.
		Lifshitz The symmetry of quasiperiodic crystals Physica A 232(3-4), 633-647 (01NOV1996) This paper describes how the Fourier-space method of crystallography is applicable to all currently-known crystallographic structures: periodic crystals, incommensurately modulated crystals, composite crystals, quasicrystals, and even modulated quasicrystals.
		While a general element of a color group is of the form (g, \gamma), where g is in the spatial point group and \gamma is a permutation of the colors, this paper concentrates on the lattice color group, having elements (e, \gamma).
	ewald.cas.usf.edu /quasibiblio.txt (4282 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.
		For example, the diagonal translation on the right is the same as the net effect of the two vertical and one horizontal basic translations shown in the middle.
	www.scienceu.com /geometry/articles/tiling/symmetry.html (812 words)

	Time as a Broken Symmetry (Site not responding. Last check: 2007-10-21)
		According to Emmy Noether’s symmetry theorem, all conservation laws of nature are derived from fundamental symmetries.
		It is also the purpose of this discussion to give some brief (but not exhausted) analyses how these three fundamental symmetries can be viewed locally as well as globally and it is hoped that some deeper differences between each perspective can be found without further rigorous discussions or the use of heavy math.
		It is also the cause of time broken symmetry that makes the structure of atom the way it is: the mass of the proton very much larger than the mass of the electron resulting in a stable atom.
	www.physicsforums.com /showthread.php?p=119588&mode=threaded (1045 words)

	Broken Symmetry
		Water has a complete rotational and translational symmetry: the pictures will look the same if the container is tipped or shoved.
		It also has a broken translational symmetry: it's easy to tell if the picture is shifted sideways, unless one shifts by a whole number of lattice units.
		If they had different symmetries, there must be a first point when mixing them when the symmetry changes, and it is usually easy to tell when that phase transition happens.
	www.lassp.cornell.edu /sethna/OrderParameters/BrokenSymmetry.html (513 words)

	An Intuitive Notion of Translations
		The word "translate" in Latin means "carried across".
		When you are sliding down a water slide, you are experiencing a translation.
		Your body is moving a given distance (the length of the slide) in a given direction.
	regentsprep.org /Regents/math/trans/Ltrans.htm (193 words)

	Internal Structure of Crystals X
		They show that when the glide translation vector becomes zero, then the faces turn out to be symmetrical with respect to the glide line g -- which consequently becomes a mirror line -- and thus to be equivalent (from a macroscopical perspective).
		After removing the glide translational difference between the pattern of Figure 6 and that of Figure 7, we can see that they are symmetrically related to each other with respect to a horizontal mirror line, expressing the point symmetry of the present Class m.
		A crystal of the same shape we already met in the Class m (Figure 16 in Part Nine), but there the crystal was the result of the combination of three Forms : one consisting of the top and bottom faces, the second consisting of the left face, and the third consisting of the right face.
	home.hetnet.nl /~heackel/d2_lattice_10.html (2450 words)

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