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Topic: Dihedral symmetry

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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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	Point groups in three dimensions - Wikipedia, the free encyclopedia
		The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself.
		In the case of multiple mirror planes and/or axes of rotation, two symmetry groups or of the same symmetry type iff there is a single rotation mapping this whole structure of the first symmetry group to that of the second.
		Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin.
	en.wikipedia.org /wiki/Point_groups_in_three_dimensions (3348 words)

	Santo to Signal: Media and Process
		The symmetry of 12 is defined by the rays in silver and etched glass and reverse gilding along the 12 sides; and the symmetry of the cross is defined by the reverse gilding of horizontal the vertical rays.
		The symmetry grouping from the first phase space portrait to the second is a transformation from the symmetry group of ornaments to the symmetry group of rosettes and to the symmetry group of frieze.
		With this symmetry covering the plane, the uniformity is broken by the use of gilding and etching techniques to conceal and reveal seven areas within the composition - seven areas that correspond to the seven sorrows of the Dolorosa.
	www.upd.edu.ph /~cfa/digitalmedia/digiteer/signal/signal3.html (2979 words)

	math lessons - Symmetry group
		The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation.
		The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing congruencies (like reflections, glide reflections and improper rotations) under which the figure is invariant.
		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.
	www.mathdaily.com /lessons/Symmetry_group (1439 words)

	BioBM 631 Lecture Oct-2-95 (Site not responding. Last check: 2007-10-10)
		Symmetry is an important concept, and undertanding the various forms of symmetry is an exercise in visualization.
		Icosahedral symmetry: has 5-fold, 3-fold, and 2-fold axes of symmetry, and is often the symmetry adopted by the protein shell of viruses (the viral capsid).
		The symmetry of the capsid extends beyond the surface of the particle, into the interior; hence, the protein subunits are wedge-shaped.
	www.mbg.cornell.edu /nicholson/biobm631/struct_levels/quaternary/quaternary.html (1605 words)

	III.C. CRYSTALS, SYMMETRY, AND DIFFRACTION (Site not responding. Last check: 2007-10-10)
		A symmetry element for which the symmetry operation is reflection across the plane combined with translation in a direction parallel to the plane.
		A symmetry operation in which each point of an object is converted to an equivalent point by projecting through a common center (called center of inversion or center of symmetry) and extending an equal distance beyond this center.
		Symmetry operations in the unit cell give rise to systematic absences in the diffraction patterns which often proves useful for determining the correct space group.
	www-ncmir.ucsd.edu /~gina/Sec-III.C.1-C.5/Sec-III.C.1-C.5.html (3590 words)

	SYMMETRY (Site not responding. Last check: 2007-10-10)
		Symmetry data defining related bond lengths, angles and dihedrals, or x, y, and z coordinates, can be included by supplying additional data after the geometry has been entered.
		The symmetry data can be the last line of the data file unless more data follows, in which case a blank line must be inserted after the symmetry data.
		Internal coordinate symmetry function 19 (see Internal Symmetry Functions) is intended for use in polymers, in which the translation vector may be a multiple of some bond-length.
	www.cachesoftware.com /mopac/Mopac2002manual/node319.html (721 words)

	Symmetry Axes
		The symmetry axes of an object are lines about which it can be rotated through some angle which brings the object to a new orientation which appears identical to its starting position.
		Some other polyhedra with the same thirteen symmetry axes are the cuboctahedron (in which the 2-fold axes pass through the vertices) and the rhombicuboctahedron (in which each axis goes through the center of two of the 26 faces).Yet another is the snub cube.
		The 2-fold axes of the cube are not axes of symmetry of the tetrahedron.
	www.georgehart.com /virtual-polyhedra/symmetry_axes.html (954 words)

	Subsymmetries (Site not responding. Last check: 2007-10-10)
		Symmetries have all sorts of relationships to one another.
		That is, the image has dihedral symmetry of order 5 but could have been made by either cyclic or dihedral symmetry.
		If a pattern with symmetry A can be produced by symmetry B, then we might call B a sub-symmetry of A.
	comp.uark.edu /~strauss/sym.2/sym.2.1.html (303 words)

	III.C. CRYSTALS, SYMMETRY, AND DIFFRACTION
		Glide Plane: A symmetry element for which the symmetry operation is reflection across the plane combined with translation in a direction parallel to the plane.
		Inversion: A symmetry operation in which each point of an object is converted to an equivalent point by projecting through a common center (called center of inversion or center of symmetry) and extending an equal distance beyond this center.
		Inversion symmetry (i): each point in the object is converted to an identical point by projecting through a common center and extending an equal distance beyond this center.
	em-outreach.ucsd.edu /web-course/Sec-III.C.1-C.5/Sec-III.C.1-C.5.html (3671 words)

	PPS '97 - Quaternary Structure - Symmetry
		Symmetry is the concept of repetitive arrangements of similar objects in space.
		Therefore, cyclic symmetries of N = 2 to infinity are permitted.
		diagram The units in a helical symmetry are related by a screw axis (a translation and rotation operation).
	www.cryst.bbk.ac.uk /pps97/course/section11/symmetry.html (644 words)

	Analysis and Synthesis in Architectural Designs by Jin-Ho Park for the Nexus Network Journal vol.3 no.1 Winter 2001 (Site not responding. Last check: 2007-10-10)
		In architectural designs, the use of symmetry may sometimes be apparent immediately by just looking at designs, although the final design is seemingly asymmetrical; or various symmetries are manifested in the parts of the designs, yet not immediately recognizable despite an almost obsessive concern for symmetry.
		Infinite symmetry groups include the motion of translation which is a shifting of the entire pattern one unit.
		The symmetry groups of the square in the third row each have two elements, while the square in the bottom row has only the identity motion in its symmetry group, that is, the no rotation less than the full turn through 360°.
	www.nexusjournal.com /Park.html (4531 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		This is due to the fact that all symmetries of the crystal are observed in the diffraction pattern, symmetries which can easily be missed in real space.
		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)

	[No title]
		Nonlinear mappings of the complex plane that possess dihedral or cylical symmetry are a natural symmetrical generalization of the Logistic map.
		To clarify the term dihedral and cyclic symmetry, we note that a square is invariant when rotated by 90, 180, 270, or 360 degrees.
		(D for the dihedral symmetry and 4 because it is a 4 sided polygon).
	chaos4.phy.ohiou.edu /~thomas/odds/symm_chaos.html (955 words)

	Dihedral Symmetry (Site not responding. Last check: 2007-10-10)
		The actual symmetry group of the surface is larger, and it is a matter of taste how large a fundamental piece one is using.
		The order 5 rotational symmetry of the surface is in this case very well visible.
		To the right is a picture of Scherk's singly periodic minimal surface with a rotational symmetry group of order 9 about the z-axes.
	www.math.uni-bonn.de /people/weber/research/minimal/notes/dihedral (217 words)

	Symmetry
		Here is another link to his superior work: Wallpaper Groups (tilings, patterns, symmetry) An undergraduate exposition on wallpaper groups, with illustrations of the 17 wallpaper groups and a wallpaper gallery.
		Using this program, I was able to find dihedral and cyclic symmetry patterns as well as a teddy bear, an evil clown, and the same clown with spider legs.
		Student projects where they wrote their names with either rotational or dihedral symmetry.
	www.mccallie.org /myates/Symmetry/symmetry.htm (1552 words)

	Möbius Deltahedra
		Each polyhedral symmetry group has a characteristic number of Möbius triangles which are connected in a tessellating pattern that covers a sphere.
		That is why you see two non-constructible cases with octahedral symmetry and two non-constructible cases with icosahedral symmetry.
		The bilateral symmetry of the equilateral triangle relative to the tetrahedral Möbius triangle implies that a new mirror plane must now be added that bisects this equilateral triangle.
	www.superliminal.com /geometry/deltahedra/deltahedra.htm (1556 words)

	Chapter 3.2
		The non-metric construction of similarity symmetry rosettes with the symmetry group K and their use in ornamental art, is based on the maintenance of the parallelism, without using the metric property of the dilatation K.
		The symmetry group L is applied in painting works having the central perspective as the element, or even as a basis of the complete central dynamic composition of the work (e.g., in the baroque, in Tintoretto's works), creating thus the visual impression of an expanding rotational motion.
		Like the antisymmetry and colored symmetry groups of rosettes, friezes and ornaments, such desymmetrizations of similarity symmetry groups are of a somewhat later date, appearing in ornamental art with dichromatic and polychromatic ceramics (in the Neolithic and in the period of the ancient civilizations).
	www.emis.de /monographs/jablan/chap32.htm (8768 words)

	The CTK Exchange Forums (Site not responding. Last check: 2007-10-10)
		At any stage, the pattern seems to have dihedral symmetry, but the centre of symmetry appears to move.
		Now, the centre of symmetry seems to be translating by the vector (1/2, 1/2) at each stage, and if we allow for that by adding it to each of the translation vectors in the rule we get the vectors (-1/2, -1/2), (-1/2, 1/2), (1/2, -1/2) and (1/2, 1/2).
		We have a rule with dihedral symmetry(This would be more easily seen in a picture than it is by working out components.
	www.cut-the-knot.org /htdocs/dcforum/DCForumID4/572.shtml (954 words)

	A Catalogue of Finite Subdivision Rules (Site not responding. Last check: 2007-10-10)
		It has rotational symmetry but not dihedral symmetry.
		The lack of dihedral symmetry makes it more challenging to show that this example is conformal.
		The Lattes subdivision rule and related examples --- gzipped ps file, pdf file --- The Lattes subdivision rule is a simple example in which the two tile types are quadrilaterals and each is subdivided into four quadrilaterals.
	www.math.vt.edu /people/floyd/catalogue (234 words)

	Molden (Site not responding. Last check: 2007-10-10)
		The third atom chosen defines the dihedral angle between the new atom, the first atom, the second atom, and the atom you are choosing.
		For dihedral angles, symmetry often dictates that a second dihedral angle be equal to the negative of the first.
		Distances, angles, and dihedral angles can be measured by clicking the appropriate option, clicking on the appropriate atoms, and looking at the small pop up window that appears.
	www.chem.ucsb.edu /~aue/chem126/molden.html (1174 words)

	Extensions, coupled cell systems and cycling chaos
		internal symmetries, one may consider two dimensional cells with dihedral symmetry or three dimensional cells with tetrahedral symmetry.
		symmetry as phenomenological models for a class of coupled cell systems which satisfy certain restrictions on coupling and individual cell dynamics.
		However, notice that we have replaced the requirement on internal symmetries by the weaker condition on invariance of hyperplanes.
	nothung.math.uh.edu /~mike/ima/node8.html (665 words)

	Research (Site not responding. Last check: 2007-10-10)
		In these works the symmetry of the cell network is important in determining the patterns of oscillation that the system can support.
		Using the lattice of isotropy subgroups, we study the normal form equations restricted to invariant fixed-point subspaces and prove that it is possible for the normal form equations to have robust, asymptotically stable, heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions.
		This feature results from the lack of certain symmetries in the cell system, which are found only in the normal forms.
	www-rohan.sdsu.edu /~antoniop/research/hcycles/dn_coupled_cells.htm (385 words)

	Structure of the stacked disk aggregate of tobacco mosaic virus protein -- Diaz-Avalos and Caspar 74 (1): 595 -- ...
		Increasing the number of images averaged (last row) reduces the noise level, enhancing the apparent dihedral symmetry of the disk pair.
		The fact that imposing the dihedral symmetry on the reference improves its correlation with the data (as in 2) indicates that
		Helical symmetry was imposed in averaging the off-equatorial layer lines, but dihedral symmetry, which would have eliminated the imaginary parts of the transform, was not imposed.
	www.biophysj.org /cgi/content/full/74/1/595 (5334 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		A pattern is generated by a single rotation through angle 2 pi/ n is called cyclic symmetry of order n and is denoted n.
		A pattern is generated by just two mirror lines meeting at angle p/n is called dihedral symmetry of order n and is denoted *n.
		Examine the sketches 5 and 5 for examples of five-fold dihedral* and cyclic symmetries.
	comp.uark.edu /~strauss/symmetry.unit/sym.1.4.html (657 words)

	LESSON 3 (Site not responding. Last check: 2007-10-10)
		A good example of this is the word 'ambulance' written reversed on the front of ambulances allowing drivers to read it in their mirrors.
		Dihedral symmetry is different from reflective symmetry because in reflective symmetry the lines of reflections do not necessarily intersect to form equal angles.
		Dihedral symmetry is different from rotational symmetry because rotational symmetry may not have lines of reflection.
	www.geom.uiuc.edu /~teach95/kt95/KTL3t.html (312 words)

	Symmetry At The Cathedral (Site not responding. Last check: 2007-10-10)
		In St. Louis, we are lucky have some of the finest mosaic artwork in North America, at the Cathedral Basilica of St. Louis.
		For a few years, Saint Louis University's MTA 124 Math and the Art of M. Escher course has punctuated their study of symmetries with a pilgrimage to the Cathedral to search for mathematical symmetries in the many mosaics.
		The cathedral has many examples of rosette symmetries, examples of all seven possible frieze symmetries, and some of the seventeen possible wallpaper symmetries.
	euler.slu.edu /Dept/Faculty/clair/cathedral (191 words)

	Illuminations: Symmetries II (Site not responding. Last check: 2007-10-10)
		The mirror line that leaves the shape unchanged is called the line of symmetry of the shape and the reflection transformation is a symmetry of the shape.
		Bilateral symmetry is very common in nature; for example, the human body and many kinds of leaves have bilateral symmetry.
		If there are 3 mirror lines, we say the figure has 3-fold dihedral symmetry or has dihedral symmetry of order 3, and so on.
	illuminations.nctm.org /index_d.aspx?id=471 (624 words)

	Uniform illumination of a sphere by N light sources
		Symmetry groups are given in OEIS A081314, raytraced visualizations for n<=21.
		For symmetric configurations the orientation is either to align an existing symmetry axis of maximal multiplicity in N-S direction or to chose y=0 as the mirror plane.
		The kind of symmetry and the order of the symmetry group of the source arrangements are indicated in the XXXXBEST.TXT files that are part of "illumtab.zip".
	www.enginemonitoring.com /illum/illum.html (4700 words)

	Cathedral Rosette Patterns (Site not responding. Last check: 2007-10-10)
		Patterns with dihedral symmetry have rotational symmetry around a central point.
		In addition to the rotational symmetry, these patterns look the same when reflected.
		One of the few examples of D5 symmetry which is not a five-pointed star embedded in another pattern.
	euler.slu.edu /~clair/cathedral/dihedral.html (67 words)

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