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| {{About}}
| | Irwin Butts is what my wife enjoys to contact me although I don't truly like being called like that. [http://www.Herpes-coldsores.com/celebrities-with-herpes.html Managing individuals] has been his [http://Www.aahealth.org/programs/comm-diseases/hiv-aids/hotlines day job] std home test for a while. Her spouse and her live in Puerto Rico but she will have to over the counter std test transfer 1 working day or another. To perform baseball is std testing at home the hobby he will by no means quit performing.<br><br>Here is my blog - [http://www.mylaffy.com/blog/view/12752/the-best-way-to-cure-an-agonizing-yeast-infection at home std test] |
| [[File:Asymmetric (PSF).svg|right|thumb|upright=0.8]]
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| [[File:Sphere symmetry group o.svg|thumb|upright=0.8|[[Sphere]] symmetrical group o representing an octahedral rotational symmetry. The yellow region shows the [[fundamental domain]].]]
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| [[File:Studio del Corpo Umano - Leonardo da Vinci.png|right|thumb|upright=0.8|[[Leonardo da Vinci]]'s ''[[Vitruvian Man]]'' (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.]]
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| [[File:BigPlatoBig.png|thumb|upright=0.8|A [[fractal]]-like shape that has [[reflectional symmetry]], [[rotational symmetry]] and [[self-similarity]], three forms of symmetry. This shape is obtained by a [[finite subdivision rule]].]]
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| [[File:Great Mosque of Kairouan, west portico of the courtyard.jpg|right|thumb|upright=0.8|Symmetric arcades of a portico in the [[Mosque of Uqba|Great Mosque of Kairouan]] also called the Mosque of Uqba, in [[Tunisia]].]]
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| '''Symmetry''' (from [[Ancient Greek|Greek]] συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement")<ref>{{cite web|title=symmetry|url=http://www.etymonline.com/index.php?term=symmetry|publisher=[[Online Etymology Dictionary]]}}</ref> has two meanings. The first is a vague sense of harmonious and beautiful proportion and balance.<ref>{{cite book|last = Zee|first = A.|title = Fearful Symmetry|publisher = Princeton University Press|location = Princeton, N.J.|year = 2007|isbn = 978-0-691-13482-6}}</ref><ref name="classical001">For example, [[Aristotle]] ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.</ref> The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a [[formal system]], such as [[geometry]] or [[physics]].
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| Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.<ref name="classical001" />
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| Mathematical symmetry may be observed
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| * with respect to the passage of [[time]];
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| * as a [[space|spatial relationship]];
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| * through geometric [[Transformation (geometry)|transformation]]s such as [[Scaling (geometry)|scaling]], [[Reflection (mathematics)|reflection]], and [[Rotation (mathematics)|rotation]];
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| * through other kinds of functional transformations;<ref>For example, operations such as moving across a regularly patterned tile floor or rotating an eight-sided [[vase]], or complex transformations of equations or in the way music is played.</ref> and
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| * as an aspect of [[abstract object]]s, [[scientific model|theoretic models]], [[language]], [[music]] and even [[knowledge]] itself.<ref name="Mainzer000">{{cite book
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| |title = Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science
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| |first = Klaus
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| |last = Mainzer
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| |publisher = World Scientific
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| |year = 2005
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| |isbn = 981-256-192-7
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| }}</ref><ref>Symmetric objects can be material, such as a person, [[crystal]], [[quilt]], [[Pamment|floor tiles]], or [[molecule]], or it can be an [[abstract object|abstract]] structure such as a [[mathematical equation]] or a series of tones ([[music]]).</ref>
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| This article describes these notions of symmetry from four perspectives. The first is symmetry in [[geometry]], which is the most familiar type of symmetry for many people. The second is the more general meaning of symmetry in [[mathematics]] as a whole. The third describes symmetry as it relates to [[science]] and [[technology]]. In this context, symmetries underlie some of the most profound results found in modern [[physics]], including aspects of [[spacetime|space and time]]. The fourth discusses symmetry in the [[humanities]], covering its rich and varied use in [[history]], [[architecture]], [[art]], and [[religion]].
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| The opposite of symmetry is [[asymmetry]].
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| ==Geometry==
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| The most familiar type of symmetry for many people is geometrical symmetry. A geometric figure (object) has ''symmetry'' if there is an [[isometry]] that maps the figure onto itself (i.e., the object has an [[Invariant (mathematics)|invariance]] under the transform). If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. For instance, a circle rotated about its center will have the same shape and size as the original circle. A circle is then said to be ''symmetric under rotation'' or to have ''rotational symmetry''. It is possible for a figure to have more than one line of symmetry.
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| The type of symmetries that are possible for a geometric object depend on the set of geometric transforms available and what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical [[group (mathematics)|group]].
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| The most common group of transforms considered is the [[Euclidean group]] of [[isometry|isometries]], or distance preserving transformations, in two dimensional ([[plane geometry]])or three dimensional ([[solid geometry]]) [[Euclidean space]]. These isometries consist of [[reflection (mathematics)|reflection]]s, [[rotation]]s, [[translation]]s and combinations of these basic operations.<ref name="Higher dimensional group theory'">[http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory]. Bangor.ac.uk. Retrieved on 2013-04-16.</ref> Under an isometric transformation, a geometric object is symmetric if the transformed object is [[Congruence (geometry)|congruent]] to the original.<ref>{{cite web|title=geometric congruence|url=http://planetmath.org/geometriccongruence|publisher=PlanetMath.org|accessdate=29 May 2013}}</ref>
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| A geometric object is typically symmetric only under a [[subgroup]] of isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups and object invariance types used in geometry.
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| ===Reflectional symmetry===
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| {{Main|reflectional symmetry}}
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| [[File:Simetria-reflexion.svg|thumb | right | 200px | An isoceles triangle with mirror symmetry. The dashed line is the axis of symmetry. Folding the triangle across the axis results in two identical halves.]]
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| [[File:Simetria-bilateria.svg|thumb | right | 200px | A drawing of a butterfly with bilateral symmetry]]
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| Reflectional symmetry, mirror symmetry, mirror-image symmetry, or [[bilateral symmetry]] is symmetry with respect to reflection.
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| In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions there is an axis of symmetry, and in three dimensions there is a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see [[mirror image]]).
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| The axis of symmetry of a two-dimensional figure is a line such that, if a [[perpendicular]] is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a [[square (geometry)|square]] has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A [[circle]] has infinitely many axes of symmetry passing through its center, for the same reason.
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| If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".
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| The [[triangle]]s with reflection symmetry are [[isosceles]], the [[quadrilateral]]s with this symmetry are the [[Kite (geometry)|kites]] and the isosceles [[trapezoid]]s.
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| For each line or plane of reflection, the [[symmetry group]] is [[isomorphic]] with C<sub>s</sub> (see [[point group]]s in three dimensions), one of the three types of order two ([[involution (mathematics)|involution]]s), hence algebraically isomorphic to C<sub>2</sub>. The [[fundamental domain]] is a [[half-plane]] or [[half-space (geometry)|half-space]].
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| === Point reflection and other involutive isometries ===
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| {{Main|Point reflection}}
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| [[File:Simetria central triangulo.png|thumb | right | 300px | Two triangles showing point reflection symmetry in the plane. Triangle ''A'B'C''' can also be generated from triangle ''ABC'' by a 180° rotation around point ''O''.]]
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| Reflection symmetry can be generalized to other [[isometry|isometries]] of {{mvar|m}}-dimensional space which are [[involution (mathematics)|involutions]], such as
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| :{{math|(''x''<sub>1</sub>, … ''x''<sub>''m''</sub>) ↦ (−''x''<sub>1</sub>, … −''x''<sub>''k''</sub>, ''x''<sub>''k''+1</sub>, … ''x''<sub>''m''</sub>)}}
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| in a certain system of [[Cartesian coordinates]]. This reflects the space along an {{math|''m''−''k''}}-dimensional [[affine subspace]].
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| If {{mvar|k}} = {{mvar|m}}, then such a transformation is known as a [[point reflection]] or an ''inversion through a point''. On the [[plane (geometry)|plane]] ({{mvar|m}} = 2) a point reflection is the same as a half-[[turn (geometry)|turn]] (180°) rotation; see below. ''Antipodal symmetry'' is an alternative name for a point reflection symmetry through the origin.<ref>{{cite book|last=tom Dieck|first=Tammo|title=Algebraic Topology|year=2008|publisher=European Mathematical Society|isbn=9783037190487|pages=261}}</ref>
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| Such a "reflection" preserves [[orientation (vector space)|orientation]] if and only if {{mvar|k}} is an [[even number|even]] number. This implies that for [[three-dimensional space|{{mvar|m}} = 3]] (as well as for other odd {{mvar|m}}) a point reflection changes the orientation of the space, like a mirror-image symmetry. That is why in physics the term ''P-[[symmetry (physics)|symmetry]]'' is used for both point reflection and mirror symmetry (P stands for [[parity (physics)|parity]]). As a point reflection in three dimensions changes a [[left-handed coordinate system]] into a [[right-handed coordinate system]], symmetry under a point reflection is also called a left-right symmetry.<ref name=Gibson1980>{{cite book|last=W.M. Gibson and B.R. Pollard|title=Symmetry principles in elementary particle physics|year=1980|publisher=Cambridge University Press|location=Cambrdge, UK|isbn=0 521 29964 0|pages=120–122}}</ref>
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| ===Rotational symmetry===
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| [[File:Simetria-rotacion.svg|right|200px|]]
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| {{Main|rotational symmetry}}
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| Rotational symmetry is symmetry with respect to some or all rotations in {{mvar|m}}-dimensional Euclidean space. Rotations are [[SE(n)|direct isometries]]; i.e., isometries preserving [[orientation (mathematics)|orientation]]. Therefore a symmetry group of rotational symmetry is a subgroup of the special Euclidean group [[SE(3)|E<sup>+</sup>({{mvar|m}})]].
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| Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E<sup>+</sup>({{mvar|m}}). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
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| For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the [[special orthogonal group]] SO({{mvar|m}}), which can be represented by the group of {{math|''m'' × ''m''}} [[Orthogonal matrix|orthogonal matrices]] with [[determinant]] 1. For {{mvar|m}} = 3 this is the [[rotation group SO(3)]].
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| In another meaning of the word, the rotation group of an object is the symmetry group within E<sup>+</sup>({{mvar|m}}), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
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| Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of [[Noether's theorem]], rotational symmetry of a physical system is equivalent to the [[angular momentum]] [[conservation law]]. See also [[rotational invariance]].
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| ===Translational symmetry===
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| [[File:Simetria-traslacional.svg|thumb| right | 200px|Congruent triangles generated by translations along the arrow.]]
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| {{Main|Translational symmetry}}
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| Translational symmetry leaves an object invariant under a discrete or continuous group of [[translation (geometry)|translations]] <math>\scriptstyle T_a(p) \;=\; p \,+\, a</math>. The illustration on the right shows four congruent triangles generated by translations along the arrow. If the line of triangles extended to infinity in both directions, they would have a discrete translational symmetry; any translation that mapped one triangle onto another would leave the whole line unchanged.
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| ===Glide reflection symmetry===
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| [[File:Simetria-antitraslacional.svg|thumb|right|200px|A glide reflection in which the upper triangle is reflected about the horizontal axis of symmetry and then translated to the right.]]
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| In the plane, a [[glide reflection]] symmetry (in 3D it is called a ''glide plane'' symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the [[frieze group]] '''p11g''' and is isomorphic with the infinite cyclic group '''Z'''.
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| ===Rotoreflection symmetry===
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| [[File:Simetria-rotoreflexion.svg|thumb|right|200px|A rotoreflection in 3D in which the upper triangle is rotated about a vertical axis and then reflected about a horizontal plane of symmetry.]]
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| In 3D, a rotoreflection or [[improper rotation]] is a rotation about an axis combined with reflection in a plane perpendicular to that axis. The symmetry groups associated with rotoreflections include:
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| * if the rotation angle has no common divisor with 360°, the symmetry group is not discrete
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| * if the rotoreflection has a 2''n''-fold rotation angle (angle of 180°/''n''), the symmetry group is ''S''<sub>2''n''</sub> of order 2''n'' (not to be confused with [[symmetric group]]s, for which the same notation is used; the abstract group is ''C<sub>2n</sub>''). A special case is ''n'' = 1, an [[Inversion in a point|inversion]], because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
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| * the group ''C<sub>nh</sub>'' (angle of 360°/''n''); for odd ''n'' this is generated by a single symmetry, and the abstract group is ''C''<sub>2''n''</sub>, for even ''n'' this is not a basic symmetry but a combination. See also [[point groups in three dimensions]].
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| ===Helical symmetry===
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| [[File:Simetria-helicoidal.svg|right|200px|]]
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| [[File:400px-Drillbit.jpg|right|thumb|400px|A drill bit with helical symmetry.]]
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| {{See also|Screw axis}}
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| [[Helix|Helical]] symmetry is the kind of symmetry seen in such everyday objects as [[spring (device)|springs]], [[Slinky]] toys, [[drill bits]], and [[auger]]s. It can be thought of as rotational symmetry along with translation along the axis of rotation, the [[screw axis]]. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an constant [[angular speed]] while simultaneously translating at a constant linear speed along its axis of rotation. At any one point in time, these two motions combine to give a ''coiling angle'' that helps define the properties of the traced helix. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the rotation is slow and the translation is speedy, the coiling angle will approach 90°.
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| Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:
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| * Infinite helical symmetry: If there are no distinguishing features along the length of a [[helix]] or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a [[Cross section (geometry)|cross section]] of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
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| *''n''-fold helical symmetry: If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called ''n-fold helical symmetry'', where ''n'' = 360°; for example, a [[double helix]]. This concept can be further generalized to include cases where <math>\scriptstyle m\theta</math> is a multiple of [[turn (geometry)|360°]] – that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
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| * Non-repeating helical symmetry: This is the case in which the angle of rotation θ required to observe the symmetry is [[irrational angle|irrational]]. The angle of rotation never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating [[Point group#In two dimensions|point group in two dimensions]]. [[DNA]], with approximately 10.5 [[base pair]]s per turn, is an example of this type of non-repeating helical symmetry.<ref>{{cite book|last=Sinden|first=Richard R.|title=DNA structure and function|year=1994|publisher=Gulf Professional Publishing|isbn=9780126457506|page=101}}</ref>
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| ===Non-isometric symmetries===
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| A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
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| *The group of [[Similarity transformation (geometry)|similarity transformation]]s; i.e., [[affine transformation]]s represented by a [[Matrix (mathematics)|matrix]] {{mvar|A}} that is a scalar times an [[orthogonal matrix]]. Thus [[homothety]] is added, [[self-similarity]] is considered a symmetry.
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| *The group of affine transformations represented by a matrix {{mvar|A}} with determinant 1 or −1; i.e., the transformations which preserve area.
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| *: This adds, e.g., oblique [[reflection symmetry]].
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| *The group of all bijective [[affine transformation]]s.
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| *The group of [[Möbius transformation]]s which preserve [[cross-ratio]]s.
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| *: This adds, e.g., [[inversive geometry|inversive]] reflections such as [[circle]] reflection on the plane.
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| In [[Felix Klein]]'s [[Erlangen program]], each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines [[Euclidean geometry]], whereas the group of Möbius transformations defines [[projective geometry]].
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| ===Scale symmetry and fractals===
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| Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does ''not'' exist for most physical systems, a point that was first discerned by [[Galileo]]. Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of [[elephant]]s versus [[mouse|mice]] (so-called [[allometric scaling]]), and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight.
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| A more subtle form of scale symmetry is demonstrated by [[fractal]]s. As conceived by [[Benoît Mandelbrot]], fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of [[magnification]] is used to examine it. A [[coast]] is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables children to use small twigs as stand-ins for full trees in [[diorama]]s, is another example.
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| This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the [[Mandelbrot set]]. Intriguingly, fractals have also found a place in CG, or [[Computer generated imagery|computer-generated movie effects]], where their ability to create very complex curves with fractal symmetries results in more realistic [[virtual world]]s.
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| ==In mathematics==
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| {{refimprove section|date=May 2013}}
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| {{Main|Symmetry in mathematics}}
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| Generalizing from geometrical symmetry in the previous section, we say that a [[mathematical object]] is ''symmetric'' with respect to a given [[Operation (mathematics)|mathematical operation]], if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a [[group (mathematics)|group]]. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and [[vice-versa|vice versa]]).
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| ===Mathematical model for symmetry===
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| The set of all symmetry operations considered on all objects in a set ''X'' can be modeled as a [[group action]] ''g'' : ''G'' × ''X'' → ''X'', where the image of ''g'' in ''G'' and ''x'' in ''X'' is written as ''g''·''x''. If, for some ''g'', ''g''·''x'' = ''y'' then ''x'' and ''y'' are said to be symmetrical to each other. For each object ''x'', operations ''g'' for which ''g''·''x'' = ''x'' form a [[Group (mathematics)|group]], the '''[[symmetry group]]''' of the object, a subgroup of ''G''. If the symmetry group of ''x'' is the trivial group then ''x'' is said to be '''asymmetric''', otherwise '''symmetric'''.
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| A general example is that ''G'' is a group of bijections ''g'': ''V'' → ''V'' acting on the set of functions ''x'': ''V'' → ''W'' by (''gx'')(''v'') = ''x''[''g''<sup>−1</sup>(''v'')] (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of ''x'' consists of all ''g'' for which ''x''(''v'') = ''x''[''g''(''v'')] for all ''v''. ''G'' is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of ''G'' may not be the symmetry group of any object. For example, if the group contains for every ''v'' and ''w'' in ''V'' a ''g'' such that ''g''(''v'') = ''w'', then only the symmetry groups of constant functions ''x'' contain that group. However, the symmetry group of constant functions is ''G'' itself.
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| In a modified version for [[vector field]]s, we have (''gx'')(''v'') = ''h''(''g'', ''x''[''g''<sup>−1</sup>(''v'')]) where ''h'' rotates any vectors and pseudovectors in ''x'', and inverts any vectors (but not pseudovectors) according to rotation and inversion in ''g'', see [[symmetry in physics]]. The symmetry group of ''x'' consists of all ''g'' for which ''x''(''v'') = ''h''(''g'', ''x''[''g''(''v'')]) for all ''v''. In this case the symmetry group of a constant function may be a proper subgroup of ''G'': a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero.
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| For a common notion of symmetry in [[Euclidean space]], ''G'' is the [[Euclidean group]] ''E''(''n''), the group of [[isometry|isometries]], and ''V'' is the Euclidean space. The '''rotation group''' of an object is the symmetry group if ''G'' is restricted to ''E''<sup>+</sup>(''n''), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions ''x'', of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (''x'' is just a [[Boolean function]] of position ''v''), or, at the other extreme; e.g., symmetry of right and left hand with all their structure.
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| For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points [[Equivalence relation|equivalent]] which, due to the symmetry, have the same properties, the [[equivalence class]]es are the [[Point stabilizer|orbits]] of the group action on the space itself. We need the value of ''x'' at one point in every orbit to define the full object. A set of such representatives forms a [[fundamental domain]]. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon [[asymmetry]].
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| An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object ''x'' we can, e.g.:
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| * Take the values in a fundamental domain (i.e., add copies of the object).
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| * Take for each orbit some kind of average or sum of the values of ''x'' at the points of the orbit (ditto, where the copies may overlap).
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| If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric.
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| As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns").
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| In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite [[cylinder (geometry)|cylinder]] with a current perpendicular to the axis; the [[magnetic field]] (a [[pseudovector]]) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the [[current density]]) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by [[angular momentum]] and [[velocity]], respectively.
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| A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} does ''not'' act transitively on all points. Equivalently, the first set is only one [[conjugacy class]] with respect to isometries, while the second has two classes.
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| ===Symmetric functions===
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| {{Main|symmetric function}}
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| A symmetric function is a function which is unchanged by any permutation of its variables. For example, ''x'' + ''y'' + ''z'' and ''xy'' + ''yz'' + ''xz'' are symmetric functions, whereas ''x''<sup>2</sup> – ''yz'' is not.
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| A function may be unchanged by a sub-group of all the permutations of its variables. For example, ''ac'' + 3''ab'' + ''bc'' is unchanged if ''a'' and ''b'' are exchanged; its symmetry group is isomorphic to C<sub>2</sub>.
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| ===In logic===
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| A [[binary relation|dyadic relation]] ''R'' is symmetric if and only if, whenever it's true that ''Rab'', it's true that ''Rba''. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul.
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| Symmetric binary [[logical connective]]s are ''[[logical conjunction|and]]'' (∧, or &), ''[[logical disjunction|or]]'' (∨, or |), ''[[logical biconditional|biconditional]]'' ([[if and only if]]) (↔), ''[[logical nand|nand]]'' (not-and, or ⊼), ''[[xor]]'' (not-biconditional, or ⊻), and ''[[logical nor|nor]]'' (not-or, or ⊽).
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| ==In science and nature==
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| {{refimprove section|date=May 2013}}
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| {{further|Patterns in nature}}
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| ===In physics===
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| {{Main|Symmetry in physics}}
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| Symmetry in physics has been generalized to mean [[Invariant (physics)|invariance]]—that is, lack of change—under any kind of transformation, for example [[General covariance|arbitrary coordinate transformations]]. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate [[Philip Warren Anderson|PW Anderson]] to write in his widely read 1972 article ''More is Different'' that "it is only slightly overstating the case to say that physics is the study of symmetry." See [[Noether's theorem]] (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; a conserved current, in Noether's original language); and also, [[Wigner's classification]], which says that the symmetries of the laws of physics determine the properties of the particles found in nature.
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| ===In physical objects===
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| ====Classical objects====
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| Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object.
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| For example, if one rotates a precisely machined aluminum [[equilateral triangle]] 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as [[optical microscope|optical]] or [[electron microscope]]s will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as [[crystal]]s or minor deformities.
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| Such simple [[thought experiment]]s show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper [[symmetry in physics#Spacetime symmetries|symmetries of space and time]] play a major role in [[classical physics]]; that is, the physics of large, everyday objects.
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| ====Quantum objects====
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| Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of [[quantum physics]], which for the most part is the physics of very small, very simple objects such as [[electron]]s, [[proton]]s, [[light]], and [[atoms]].
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| Unlike everyday objects, objects such as [[electron]]s have very limited numbers of configurations, called [[Quantum state|state]]s, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an [[observation|observer]] is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion ''F''(''x'') = ''x'' ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.
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| ====Consequences of quantum symmetry====
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| While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.
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| However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by [[Richard Feynman]] in the direct transcripts of his [[Feynman Lectures on Physics]], Volume III, Section 3.4, ''Identical particles''. (Unfortunately, the quote was edited out of the printed version of the same lecture.)
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| {{quote|… if there is a physical situation in which it is impossible to tell which way it happened, it ''always'' interferes; it ''never'' fails.}}
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| The word "[[quantum interference|interferes]]" in this context is a quick way of saying that such objects fall under the rules of [[quantum mechanics]], in which they behave more like [[wave]]s that interfere than like everyday large objects.
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| In short, when an object becomes so simple that a symmetry assertion of the form ''F(x) = x'' becomes an exact statement of experimentally verifiable sameness, ''x'' ceases to follow the rules of [[classical physics]] and must instead be modeled using the more complex, and often far less intuitive, rules of [[quantum physics]].
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| This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.
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| ===Generalizations of symmetry===
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| If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of [[symmetry group]] to that of a [[groupoid]]. Indeed, A. [[Connes]] in his book "[[Non-commutative geometry]]" writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum.
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| The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of ''higher order symmetry'' which have been a little explored, as follows.
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| The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group ''G'' naturally form a [[crossed module]] <math>\scriptstyle G \;\to\; \mathrm{Aut}(G)</math>, and crossed modules give an algebraic model of homotopy 2-types. At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is known to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed ''n''-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics.<ref name="Higher dimensional group theory'"/><ref>[http://golem.ph.utexas.edu/category/ n-category cafe] – discussion of ''n''-groups</ref>
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| Physicists have come up with other directions of generalization, such as [[supersymmetry]] and [[quantum group]]s, yet the different options are indistinguishable during various circumstances.
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| [[File:Chance and a Half, Posing.jpg|thumb|upright|Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.]]
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| ===In biology===
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| {{Further|symmetry in biology|facial symmetry|patterns in nature}}
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| [[Bilateria|Bilateral animals]], including humans, are more or less symmetric with respect to the [[sagittal plane]] which divides the body into left and right halves.<ref>{{cite web |last=Valentine |first=James W. |title=Bilateria |url=http://www.accessscience.com/abstract.aspx?id=802620&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fid%3d802620 |publisher=AccessScience |accessdate=29 May 2013}}</ref> Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The [[cephalisation|head becomes specialized]] with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.<ref>{{cite web | url=http://biocongroup.eu/DA/Calendario_files/Bilateria.pdf | title=Animal Diversity (Third Edition) | publisher=McGraw-Hill | work=Chapter 8: Acoelomate Bilateral Animals | year=2002 | accessdate=October 25, 2012 | author=Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan | pages=139}}</ref>
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| Plants and sessile (attached) animals such as [[sea anemone]]s often have radial or [[rotational symmetry]], which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the [[echinoderms]], the group that includes [[starfish]], [[sea urchin]]s, and [[sea lilies]].<ref>{{cite book | title=What Shape is a Snowflake? Magical Numbers in Nature | publisher=Weidenfeld & Nicolson | author=Stewart, Ian | year=2001 | pages=64–65}}</ref>
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| ===In chemistry===
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| {{Main|molecular symmetry}}
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| Symmetry is important to [[chemistry]] because it explains observations in [[spectroscopy]], [[quantum chemistry]] and [[crystallography]]. It draws heavily on [[group theory]].
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| ==In history, religion, and culture==
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| ===In social interactions===
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| People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you." Peer relationships are based on symmetry, power relationships are based on asymmetry.<ref>[http://www.emotionalcompetency.com/symmetry.htm Emotional Competency] Entry describing Symmetry</ref>
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| ===In architecture===
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| [[File:Isfahan Lotfollah mosque ceiling symmetric.jpg|right|thumb|The ceiling of [[Lotfollah mosque]], [[Isfahan]], [[Iran]] has rotational symmetry of order eight and eight lines of reflection.]]
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| [[File:Lightmatter pisa.jpg|thumb|left|upright|Leaning Tower of Pisa]]
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| [[File:Taj Mahal, Agra views from around (85).JPG|right|thumb|The Taj Mahal has bilateral symmetry.]]
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| Architects have long employed overall symmetry as a method of telling the viewer that a building is intended to convey an important idea, that the building is a physical expression of an important belief and ideals of the people who commissioned it. Symmetry endows a formal quality that distinguishes itself from the asymmetrical forms that are more casual. Such a building with an expressive intention is therefore, by definition, monumental, as it seeks to memorialize an idea. Symmetry is an expressive device, often used in conjunction with other methods like magnified size, elevation above ground level, and use of permanent materials. All are ways of communicating the importance and validity of the idea that is intended to be memorialized.
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| Examples of ancient architectures that made powerful use of symmetry to convey importance included the [[Egypt]]ian [[Pyramids]], the [[Greece|Greek]] [[Parthenon]], the first and second [[Temple of Jerusalem]], China's [[Forbidden City]], [[Cambodia]]'s [[Angkor Wat]] complex, and the many temples and pyramids of ancient [[Pre-Columbian]] civilizations. More recent historical examples of architectures emphasizing symmetries include [[Gothic architecture]] cathedrals, and [[United States|American]] President [[Thomas Jefferson]]'s [[Monticello]] home. The [[Taj Mahal]] is also an example of symmetry.<ref>{{cite book|author=Derry, Gregory N. |title=What Science Is and How It Works |url=http://books.google.com/books?id=Dk-xS6nABrYC&pg=PA269 |year=2002 |publisher=Princeton University Press |isbn=978-1-4008-2311-6 |pages=269–}}</ref>
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| An example of a [[broken symmetry]] in architecture is the [[Leaning Tower of Pisa]], whose notoriety stems not from the intended symmetry of its design, but from the violation of that symmetry from the lean that developed while it was still under construction. It conveys the failure of the idea it sought to memorialize. Modern examples of architectures that make impressive or complex use of various symmetries include [[Australia]]'s [[Sydney Opera House]] and [[Houston, Texas]]'s simpler [[Astrodome]].
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| Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual [[floor plan]]s, and down to the design of individual building elements such as intricately carved doors, [[stained glass window]]s, [[mosaic|tile mosaics]], [[frieze]]s, stairwells, stair rails, and [[balustrades]]. For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, [[Islam]]ic buildings such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the general prohibition of Islam against using images of people or animals.<ref>[http://members.tripod.com/vismath/kim/ Williams: Symmetry in Architecture]. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.</ref><ref>[http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Aslaksen: Mathematics in Art and Architecture]. Math.nus.edu.sg. Retrieved on 2013-04-16.</ref>[[Modernist architecture]], starting with [[International style (architecture)|International style]], sometimes rejects symmetry, relying on wings and balance of masses.{{citation needed|date=May 2013}}
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| ===In pottery and metal vessels===
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| [[File:Makingpottery.jpg|thumb|Clay pots thrown on a [[pottery wheel]] acquire rotational symmetry.]]
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| Since the earliest uses of [[pottery wheel]]s to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives. For example, [[Persia]]n pottery dating from the fourth millennium BC and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce visually striking overall designs.{{citation needed|date=May 2013}}
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| Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient [[Chinese people|Chinese]], for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.<ref>[http://www.chinavoc.com/arts/handicraft/bronze.htm The Art of Chinese Bronzes]. Chinavoc (2007-11-19). Retrieved on 2013-04-16.</ref><ref>[http://www-oi.uchicago.edu/OI/MUS/VOL/NN_SUM94/NN_Sum94.html Grant: Iranian Pottery in the Oriental Institute]. Oi.uchicago.edu (2007-07-30). Retrieved on 2013-04-16.</ref><ref>[http://web.archive.org/web/20040203213334/http://www.metmuseum.org/collections/department.asp?dep=14 Introduction to Islamic Art]. The Metropolitan Museum of Art, New York.</ref>
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| ===In quilts===
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| [[File:kitchen kaleid.svg|thumb|120px|left|Kitchen Kaleidoscope Block]]
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| As [[quilt]]s are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.<ref>[http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm Quate: Exploring Geometry Through Quilts]. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.</ref>
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| ===In carpets and rugs===
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| [[File:Farsh1.jpg|thumb|300px|right|Persian rug.]]
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| A long tradition of the use of symmetry in [[carpet]] and rug patterns spans a variety of cultures. American [[Navajo people|Navajo]] Indians used bold diagonals and rectangular motifs. Many [[Oriental rugs]] have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, [[Motif (visual arts)|motifs]] that are reflected across both the horizontal and vertical axes.<ref>[http://web.archive.org/web/20010203155200/http://marlamallett.com/default.htm Marla Mallett Textiles & Tribal Oriental Rugs]. The Metropolitan Museum of Art, New York.</ref><ref>[http://navajocentral.org/rugs.htm Dilucchio: Navajo Rugs]. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.</ref>
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| ===In music===
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| <imagemap>
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| File:Major and minor triads.png|thumb|right|<span style="color:red;">Major</span> and <span style="color:blue;">minor</span> triads on the white piano keys are symmetrical to the D. [[Major and minor#Major and minor scales|(compare article)]] <small>[[:File:Major and minor triads.png|<span style="color:#aaa;">(file)</span>]]</small>
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| poly 35 442 35 544 179 493 [[A minor|root of A minor triad]]
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| poly 479 462 446 493 479 526 513 492 [[A minor|third of A minor triad]]
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| poly 841 472 782 493 840 514 821 494 [[A minor|fifth of A minor triad]]
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| poly 926 442 875 460 906 493 873 525 926 545 [[A minor|fifth of A minor triad]]
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| poly 417 442 417 544 468 525 437 493 469 459 [[C major|root of C major triad]]
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| poly 502 472 522 493 502 514 560 493 [[C major|root of C major triad]]
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| poly 863 462 830 493 863 525 895 493 [[C major|third of C major triad]]
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| poly 1303 442 1160 493 1304 544 [[C major|fifth of C major triad]]
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| poly 280 406 264 413 282 419 275 413 [[E minor|fifth of E minor triad]]
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| poly 308 397 293 403 301 412 294 423 309 428 [[E minor|fifth of E minor triad]]
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| poly 844 397 844 428 886 413 [[E minor|root of E minor triad]]
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| poly 1240 404 1230 412 1239 422 1250 412 [[E minor|third of E minor triad]]
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| poly 289 404 279 413 288 422 300 413 [[G major|third of G major triad]]
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| poly 689 398 646 413 689 429 [[G major|fifth of G major triad]]
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| poly 1221 397 1222 429 1237 423 1228 414 1237 403 [[G major|root of G major triad]]
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| poly 1249 406 1254 413 1249 418 1265 413 [[G major|root of G major triad]]
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| poly 89 567 73 573 90 579 86 573 [[D minor|fifth of D minor triad]]
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| poly 117 558 102 563 111 572 102 583 118 589 [[D minor|fifth of D minor triad]]
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| poly 650 558 650 589 693 573 [[D minor|root of D minor triad]]
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| poly 1050 563 1040 574 1050 582 1061 574 [[D minor|third of D minor triad]]
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| poly 98 565 88 573 98 583 110 574 [[F major|third of F major triad]]
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| poly 498 558 455 573 498 589 [[F major|fifth of F major triad]]
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| poly 1031 557 1031 589 1047 583 1038 574 1046 563 [[F major|root of F major triad]]
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| poly 1075 573 1059 580 1064 573 1058 567 [[F major|root of F major triad]]
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| desc none
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| </imagemap>
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| Symmetry is not restricted to the visual arts. Its role in the history of [[music]] touches many aspects of the creation and perception of music.
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| ====Musical form====
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| Symmetry has been used as a [[musical form|formal]] constraint by many composers, such as the [[arch form|arch (swell) form]] (ABCBA) used by [[Steve Reich]], [[Béla Bartók]], and [[James Tenney]]. In classical music, Bach used the symmetry concepts of permutation and invariance.<ref>see ("Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave])</ref>
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| ====Pitch structures====
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| Symmetry is also an important consideration in the formation of [[scale (music)|scale]]s and [[chord (music)|chords]], traditional or [[tonality|tonal]] music being made up of non-symmetrical groups of [[pitch (music)|pitches]], such as the [[diatonic scale]] or the [[major chord]]. [[Symmetrical scale]]s or chords, such as the [[whole tone scale]], [[augmented chord]], or diminished [[seventh chord]] (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are [[ambiguous]] as to the [[Key (music)|key]] or tonal center, and have a less specific [[diatonic functionality]]. However, composers such as [[Alban Berg]], [[Béla Bartók]], and [[George Perle]] have used axes of symmetry and/or [[interval cycle]]s in an analogous way to [[musical key|keys]] or non-[[tonality|tonal]] tonal [[Tonic (music)|center]]s.
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| Perle (1992){{clarify|date=May 2013}} explains "C–E, D–F♯, [and] Eb–G, are different instances of the same [[interval (music)|interval]] … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"
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| {|
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| |-
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| |D
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| |D♯
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| |'''E'''
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| |F
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| |F♯
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| |G
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| |G♯
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| |-
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| |D
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| |C♯
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| |'''C'''
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| |B
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| |A♯
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| |A
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| |G♯
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| |}
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| Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).
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| {|
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| |rowspan=3|+
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| |2
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| |3
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| |'''4'''
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| |5
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| |6
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| |7
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| |8
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| |-
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| |2
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| |1
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| |'''0'''
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| |11
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| |10
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| |9
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| |8
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| |-
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| |4
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| |4
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| |4
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| |4
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| |4
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| |4
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| |4
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| |}
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| Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are [[enharmonic]] with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal [[chord progression|progressions]] in the works of [[Romantic music|Romantic]] composers such as [[Gustav Mahler]] and [[Richard Wagner]] form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, [[Alexander Scriabin]], [[Edgard Varèse]], and the Vienna school. At the same time, these progressions signal the end of tonality.
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| The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's ''Quartet'', Op. 3 (1910).<ref>{{cite book | title=The Listening Composer | publisher=University of California Press | author=Perle, George | year=1990}}</ref>
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| ====Equivalency====
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| [[Tone row]]s or [[pitch class]] [[Set theory (music)|sets]] which are [[Invariant (music)|invariant]] under [[Permutation (music)|retrograde]] are horizontally symmetrical, under [[inversion (music)|inversion]] vertically. See also [[Asymmetric rhythm]].
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| ===In other arts and crafts===
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| [[File:Celticknotwork.png|frame|[[Celtic knot]]work]]
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| Symmetries appear in the design of objects of all kinds. Examples include [[beadwork]], [[furniture]], [[sand painting]]s, [[knot]]work, [[masks]], and [[musical instruments]]. Symmetries are central to the art of [[M.C. Escher]] and the many applications of [[tessellation|tiling]].
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| ===In aesthetics===
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| {{Main|Symmetry (physical attractiveness)}}
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| The relationship of symmetry to [[aesthetics]] is complex. Humans use [[bilateral symmetry]] to judge the likely health or fitness of other people.{{citation needed|date=May 2013}} Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. People prefer shapes that have some symmetry, but enough complexity to make them interesting.<ref>{{cite book|last = Arnheim|first = Rudolf|title = Visual Thinking|publisher = University of California Press|year = 1969}}</ref>
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| ==See also==
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| {{colbegin|2}}
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| '''Symmetry in [[statistics]]'''
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| *[[Skewness]], asymmetry of a statistical distribution
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| '''Symmetry in games and puzzles'''
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| *[[Symmetric games]]
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| *[[Sudoku]]
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| '''Symmetry in literature'''
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| *[[Palindrome]]
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| '''Moral symmetry'''
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| *[[Empathy]] and [[Sympathy]]
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| *[[Golden Rule]]
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| *[[Reciprocity (social psychology)|Reciprocity]]
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| *[[Reflective equilibrium]]
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| *[[Tit for tat]]
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| '''Other'''
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| *[[Asymmetric rhythm]]
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| *[[Asymmetry]]
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| *[[Burnside's lemma]]
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| *[[Chirality (mathematics)|Chirality]]
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| *[[M.C. Escher]]
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| *[[Even and odd functions]]
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| *[[Fixed points of isometry groups in Euclidean space]] – center of symmetry
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| *[[Gödel, Escher, Bach]]
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| *[[Ignacio Matte Blanco]]
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| *[[Semimetric]], which is sometimes translated as symmetric in Russian texts.
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| *[[Spacetime symmetries]]
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| *[[Spontaneous symmetry breaking]]
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| *[[Symmetric relation]]
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| *[[Polyiamond#Symmetries|Symmetries of polyiamonds]]
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| *[[Free polyomino|Symmetries of polyominoes]]
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| *[[Symmetry group]]
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| *[[T-symmetry|Time symmetry]]
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| *[[Wallpaper group]]
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| {{colend}}
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| ==References==
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| {{reflist|26em}}
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| ==Further reading==
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| * ''The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry'', [[Mario Livio]], [[Souvenir Press]] 2006, ISBN 0-285-63743-6
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| ==External links==
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| {{Wiktionary}}
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| {{Commons category|Symmetry}}
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| *[http://www.teachersnetwork.org/teachnet/westchester/symmetry.htm Calotta: A World of Symmetry]
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| *[http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane]
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| *[http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
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| *[http://www.mi.sanu.ac.rs/~jablans/isis0.htm ISIS Symmetry]
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| *[http://www.symmetry.hu/isa_articles.html ''International Symmetry Association – ISA'']
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| *Institute [http://www.symmetry.hu/aus_symmetrion.html ''Symmetrion'']
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| [[Category:Symmetry|*]]
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