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[[Vaizdas:Asymmetric (PSF).svg|right|thumb|upright=0.8|Veidrodinės simetrijos ir asimetrijos pavyzdžiai]]
[[Vaizdas:Sphere symmetry group o.svg|thumb|upright=0.8|[[Sfera|Sferinės]] simetrijos grupė '''O''' atitinka oktaedrinę sukimo simetriją. Geltonai pažymėta ''fundamentalioji sritis'', kurioje yra visi atitinkamų simetrinių pokyčių metu sutampantys taškai.]]
[[Vaizdas:Studio del Corpo Umano - Leonardo da Vinci.png|right|thumb|upright=0.8|[[Leonardo da Vinci|Leonardo da Vinčio]] '[[Vitruvijaus žmogus]]' (apie 1487) dažnai naudojamas kaip žmogaus kūno, o kartu ir visos gamtos, simetriškumo simbolis.]]
[[Vaizdas:BigPlatoBig.png|thumb|upright=0.8|[[Fraktalas|Fraktalinis]] pavidalas, pasižymintis '''atspindžio simetrija''', '''sukimo simetrija''' ir '''savipanašumu''', trimis simetrijos formomis. Šis pavidalas gautas, taikant ''baigtinio dalijimo taisyklę''.]]
[[Vaizdas:Great Mosque of Kairouan, west portico of the courtyard.jpg|right|thumb|upright=0.8|Simetriškos arkados Didžiosios Kairuano (arba Ukbos) mečetės portike, [[Tunisas|Tunise]].]]
'''Simetrija''' ({{gr|συμμετρία}} ''symmetria'' – „vienodai matuojamas, proporcingas, suderintas“)<ref>{{cite web |title=symmetry |url=http://www.etymonline.com/index.php?term=symmetry |publisher=[[Online Etymology Dictionary]]}}</ref> kas dienėje kalboje reiškia harmoniją, darnumą ir grožį reiškiančią proporciją ir suderinimą, dalių pusiausvyrą. Pavyzdžiui [[Aristotelis]] sferos pavidalą suteikė dangaus kūnams, taip suteikdamas geometrinei simetrijos prasmei natūralios tvarkos pobūdį ir, tuo pačiu, siekdamas parodyti kosmoso tobulumą. Matematikoje ir kituose moksluose '''simetrija''' apibrėžiama griežčiau – tai objekto savybė, kai pokyčio (atspindėjimo, ar kitokios simetrinės transformacijos) metu jis gali išlikti [[Invariantas|invariantiškas]] (nepakitęs). Nors kasdienės ir mokslinės '''simetrijos''' prasmės gali atrodyti nutolusios viena nuo kitos, iš tiesų, jos yra susijusios, todėl galima jas nagrinėti kartu. Taip, pavyzdžiui, kūno ''sferinė simetrija'' reiškia, kad kūną, kuris pasižymi tokia simetrija, pasukant erdvėje aplink nejudantį tašką, jo forma ir dydis nepasikeičia. ''Atspindžio simetrija'' reiškia, kad dešinioji ir kairioji objekto pusės atrodo visiškai vienodai. Gamtoje dažnai aptinkama [[biologinė simetrija]] yra apytikslė simetrija.
Mathematical symmetry may be observed with respect to the passage of [[time]]; as a [[space|spatial relationship]]; through [[geometric transformation]]s such as [[Scaling (geometry)|scaling]], [[Reflection (mathematics)|reflection]], and [[Rotation (mathematics)|rotation]]; through other kinds of functional transformations; and as an aspect of [[abstract object]]s, [[scientific model|theoretic models]], [[language]], [[music]] and even [[knowledge]] itself.<ref name="Mainzer000">{{cite book |title = Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science |first = Klaus |last = Mainzer |publisher = World Scientific |year = 2005 |isbn = 981-256-192-7}}</ref>{{efn|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]]).}}
This article describes symmetry from three perspectives: in [[mathematics]], including [[geometry]], the most familiar type of symmetry for many people; in [[science]] and [[nature]]; and in the arts, covering [[architecture]], [[art]] and [[music]].
The opposite of symmetry is [[asymmetry]].
==In mathematics==
===In geometry===
{{main|Symmetry (geometry)}}
[[File:The armoured triskelion on the flag of the Isle of Man.svg|thumb|upright=0.6|The [[triskelion]] has 3-fold rotational symmetry.]]
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.<ref> E. H. Lockwood, R. H. Macmillan, ''Geometric Symmetry'', London: Cambridge Press,
1978</ref> This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
* An object has [[reflectional symmetry]] if there is a line of symmetry going through it which divides it into two pieces which are mirror images of each other.<ref>{{cite book |title=Symmetry |last=Weyl |first=Hermann |authorlink=Hermann Weyl |year=1982 |origyear=1952 |publisher=Princeton University Press |location=Princeton | isbn=0-691-02374-3 |pages= |url= |ref=Weyl 1982}}</ref>
*An object has [[rotational symmetry]] if the object can be rotated about a fixed point without changing the overall shape.<ref>{{cite book | author=Singer, David A. | year=1998 | title=Geometry: Plane and Fancy | publisher=Springer Science & Business Media}}</ref>
*An object has [[translational symmetry]] if it can be [[translation (geometry)|translated]] without changing its overall shape.<ref>Stenger, Victor J. (2000) and Mahou Shiro (2007). ''Timeless Reality''. Prometheus Books. Especially chapter 12. Nontechnical.</ref>
*An object has [[helical symmetry]] if it can be simultaneously translated and rotated in three-dimensional space along a line known as a [[screw axis]].<ref>Bottema, O, and B. Roth, ''Theoretical Kinematics,'' Dover Publications (September 1990)</ref>
*An object has [[scale symmetry]] if it does not change shape when it is expanded or contracted.<ref>Tian Yu Cao ''Conceptual Foundations of Quantum Field Theory'' Cambridge University Press p.154-155</ref> [[Fractals]] also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions.<ref name="Gouyet">{{cite book | last = Gouyet | first = Jean-François | title = Physics and fractal structures | publisher = Masson Springer | location = Paris/New York | year = 1996 | isbn = 978-0-387-94153-0 }}</ref>
*Other symmetries include [[glide reflection]] symmetry and [[improper rotation|rotoreflection]] symmetry.
===In logic===
A [[binary relation|dyadic relation]] ''R'' is symmetric if and only if, whenever it's true that ''Rab'', it's true that ''Rba''.<ref>Josiah Royce, Ignas K. Skrupskelis (2005) ''The Basic Writings of Josiah Royce: Logic, loyalty, and community (Google eBook)'' Fordham Univ Press, p. 790</ref> 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.
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 ⊽).
===Other areas of mathematics===
{{main|Symmetry (mathematics)}}
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.<ref>Christopher G. Morris
(1992) ''Academic Press Dictionary of Science and Technology'' Gulf Professional Publishing</ref> The set of operations that preserve a given property of the object form a [[group (mathematics)|group]].
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include [[even and odd functions]] in [[calculus]]; the [[symmetric group]] in [[abstract algebra]]; [[symmetric matrix|symmetric matrices]] in [[linear algebra]]; and the [[Galois group]] in [[Galois theory]]. In [[statistics]], it appears as [[symmetric probability distribution]]s, and as [[skewness]], asymmetry of distributions.{{cn|date=May 2015}}
==In science and nature==
{{further|Patterns in nature}}
===In physics===
{{Main|Symmetry in physics}}
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]].<ref>{{cite book |title = Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries |first1 = Giovanni |last1 = Costa |first2=Gianluigi |last2=Fogli| publisher = Springer Science & Business Media |year = 2012 |pages = 112}}</ref> 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."<ref>{{cite journal | last=Anderson | first=P.W. | title=More is Different | journal=[[Science (journal)|Science]] | volume=177 | issue=4047| pages=393–396 | year=1972 | url=http://robotics.cs.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf | doi=10.1126/science.177.4047.393 | pmid=17796623 | format=|bibcode = 1972Sci...177..393A }}</ref> 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);<ref name=Noether>{{Cite book | last = Kosmann-Schwarzbach | first = Yvette | authorlink = Yvette Kosmann-Schwarzbach | title = The Noether theorems: Invariance and conservation laws in the twentieth century | publisher = [[Springer Science+Business Media|Springer-Verlag]] | series = Sources and Studies in the History of Mathematics and Physical Sciences | year = 2010 | isbn = 978-0-387-87867-6}}</ref> and also, [[Wigner's classification]], which says that the symmetries of the laws of physics determine the properties of the particles found in nature.<ref>{{citation|first=E. P.|last=Wigner|authorlink=Eugene Wigner|title=On unitary representations of the inhomogeneous Lorentz group|journal=[[Annals of Mathematics]]|issue=1|volume=40|pages=149–204|year=1939|doi=10.2307/1968551|mr=1503456 }}.</ref>
Important symmetries in physics include [[continuous symmetry|continuous symmetries]] and [[discrete symmetry|discrete symmetries]] of [[spacetime]]; [[internal symmetry|internal symmetries]] of particles; and [[supersymmetry]] of physical theories.
[[File:Chance and a Half, Posing.jpg|thumb|upright|Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.]]
===In biology===
{{Further|symmetry in biology|facial symmetry|patterns in nature}}
[[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>
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>
===In chemistry===
{{Main|molecular symmetry}}
Symmetry is important to [[chemistry]] because it undergirds essentially all ''specific'' interactions between molecules in nature (i.e., via the interaction of natural and human-made [[chiral (chemistry)|chiral]] molecules with inherently chiral biological systems). The control of the [[molecular symmetry|symmetry]] of molecules produced in modern [[chemical synthesis]] contributes to the ability of scientists to offer [[drug|therapeutic]] interventions with minimal [[side effects]]. A rigorous understanding of symmetry explains fundamental observations in [[quantum chemistry]], and in the applied areas of [[spectroscopy]] and [[crystallography]]. The theory and application of symmetry to these areas of [[physical science]] draws heavily on the mathematical area of [[group theory]].<ref>{{cite book | author=Lowe, John P; Peterson, Kirk | title=Quantum Chemistry | publisher=Academic Press| edition=Third | year=2005 | isbn=0-12-457551-X}}</ref>
==In social interactions==
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of [[Reciprocity (social psychology)|Reciprocity]], [[empathy]], [[sympathy]], [[apology]], [[dialog]], respect, [[justice]], and [[revenge]].
[[Reflective equilibrium]] is the balance that may be attained through deliberative mutual adjustment among general principles and specific [[judgment]]s.<ref>{{sep entry|reflective-equilibrium|Reflective Equilibrium|[[Norman Daniels]]|2003-04-28}}</ref>
Symmetrical interactions send the [[morality|moral]] message "we are all the same" while asymmetrical interactions may send the message "I am special; better than you." Peer relationships, such as can be governed by the [[golden rule]], are based on symmetry, whereas power relationships are based on asymmetry.<ref>[http://www.emotionalcompetency.com/symmetry.htm Emotional Competency]: Symmetry</ref> Symmetrical relationships can to some degree be maintained by simple ([[game theory]]) strategies seen in [[symmetric games]] such as [[tit for tat]].<ref>{{cite web|last1=Lutus|first1=P.|title=The Symmetry Principle|url=http://www.arachnoid.com/symmetry/details.html|accessdate=28 September 2015|date=2008}}</ref>
==In the arts==
[[File:Isfahan Lotfollah mosque ceiling symmetric.jpg||thumb|The ceiling of [[Lotfollah mosque]], [[Isfahan]], [[Iran]] has 8-fold symmetries.]]
{{further|Mathematics and art}}
===In architecture===
{{further|Mathematics and architecture}}
[[File:Taj Mahal, Agra views from around (85).JPG|thumb|Seen from the side, the [[Taj Mahal]] has bilateral symmetry; from the top (in plan), it has fourfold symmetry.]]
Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic [[cathedral]]s and [[The White House]], through the layout of the individual [[floor plan]]s, and down to the design of individual building elements such as [[mosaic|tile mosaics]]. [[Islam]]ic buildings such as the [[Taj Mahal]] and the [[Lotfollah mosque]] make elaborate use of symmetry both in their structure and in their ornamentation.<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> Moorish buildings like the [[Alhambra]] are ornamented with complex patterns made using translational and reflection symmetries as well as rotations.<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>
It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";<ref name=Dunlap>{{cite web |last1=Dunlap |first1=David W. |title=Behind the Scenes: Edgar Martins Speaks |url=http://lens.blogs.nytimes.com/2009/07/31/behind-10/?_r=0 |publisher=New York Times |accessdate=11 November 2014 |date=31 July 2009 | quote=“My starting point for this construction was a simple statement which I once read (and which does not necessarily reflect my personal views): ‘Only a bad architect relies on symmetry; instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses.’}}</ref> [[Modernist architecture]], starting with [[International style (architecture)|International style]], relies instead on "wings and balance of masses".<ref name=Dunlap/>
===In pottery and metal vessels===
[[File:Makingpottery.jpg|thumb|left|upright=0.7|Clay pots thrown on a [[pottery wheel]] acquire rotational symmetry.]]
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.
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>
===In quilts===
[[File:kitchen kaleid.svg|thumb|120px|left|Kitchen Kaleidoscope Block]]
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>
===In carpets and rugs===
[[File:Farsh1.jpg|thumb|300px|right|Persian rug.]]
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>
===In music===
<imagemap>
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>
poly 35 442 35 544 179 493 [[A minor|root of A minor triad]]
poly 479 462 446 493 479 526 513 492 [[A minor|third of A minor triad]]
poly 841 472 782 493 840 514 821 494 [[A minor|fifth of A minor triad]]
poly 926 442 875 460 906 493 873 525 926 545 [[A minor|fifth of A minor triad]]
poly 417 442 417 544 468 525 437 493 469 459 [[C major|root of C major triad]]
poly 502 472 522 493 502 514 560 493 [[C major|root of C major triad]]
poly 863 462 830 493 863 525 895 493 [[C major|third of C major triad]]
poly 1303 442 1160 493 1304 544 [[C major|fifth of C major triad]]
poly 280 406 264 413 282 419 275 413 [[E minor|fifth of E minor triad]]
poly 308 397 293 403 301 412 294 423 309 428 [[E minor|fifth of E minor triad]]
poly 844 397 844 428 886 413 [[E minor|root of E minor triad]]
poly 1240 404 1230 412 1239 422 1250 412 [[E minor|third of E minor triad]]
poly 289 404 279 413 288 422 300 413 [[G major|third of G major triad]]
poly 689 398 646 413 689 429 [[G major|fifth of G major triad]]
poly 1221 397 1222 429 1237 423 1228 414 1237 403 [[G major|root of G major triad]]
poly 1249 406 1254 413 1249 418 1265 413 [[G major|root of G major triad]]
poly 89 567 73 573 90 579 86 573 [[D minor|fifth of D minor triad]]
poly 117 558 102 563 111 572 102 583 118 589 [[D minor|fifth of D minor triad]]
poly 650 558 650 589 693 573 [[D minor|root of D minor triad]]
poly 1050 563 1040 574 1050 582 1061 574 [[D minor|third of D minor triad]]
poly 98 565 88 573 98 583 110 574 [[F major|third of F major triad]]
poly 498 558 455 573 498 589 [[F major|fifth of F major triad]]
poly 1031 557 1031 589 1047 583 1038 574 1046 563 [[F major|root of F major triad]]
poly 1075 573 1059 580 1064 573 1058 567 [[F major|root of F major triad]]
desc none
</imagemap>
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.
====Musical form====
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>
====Pitch structures====
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.
{{harvtxt|Perle|1992}}<ref>{{Cite journal |title=Symmetry, the twelve-tone scale, and tonality |first=George |last=Perle |authorlink=George Perle |journal=Contemporary Music Review |volume=6 |issue=2 |year=1992 |pages=81–96 |doi=10.1080/07494469200640151}}</ref> 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:"
{|
|-
|D
|
|D♯
|
|'''E'''
|
|F
|
|F♯
|
|G
|
|G♯
|-
|D
|
|C♯
|
|'''C'''
|
|B
|
|A♯
|
|A
|
|G♯
|}
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).
{|
|rowspan=3|+
|2
|
|3
|
|'''4'''
|
|5
|
|6
|
|7
|
|8
|-
|2
|
|1
|
|'''0'''
|
|11
|
|10
|
|9
|
|8
|-
|4
|
|4
|
|
|
|4
|
|4
|
|4
|
|4
|}
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.
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>
====Equivalency====
[[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]].
===In other arts and crafts===
[[File:Celticknotwork.png|frame|[[Celtic knot]]work]]
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]] in art and craft forms such as [[wallpaper]], ceramic tilework, [[batik]], [[ikat]], carpet-making, and many kinds of [[textile]] and [[embroidery]] patterns.<ref>{{cite book |last1=Cucker |first1=Felix |title=Manifold Mirrors: The Crossing Paths of the Arts and Mathematics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-72876-8 |pages=77–78, 83, 89, 103}}</ref>
===In aesthetics===
{{Main|Symmetry (physical attractiveness)}}
The relationship of symmetry to [[aesthetics]] is complex. Humans find [[bilateral symmetry]] in faces physically attractive;<ref name="Grammer1994">Grammer, K., & Thornhill, R. (1994). Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness. Journal of comparative psychology (Washington, D.C. : 1983), 108(3), 233–42.</ref> it indicates health and genetic fitness.<ref>{{cite book | last = Rhodes | first = Gillian | coauthors = Zebrowitz, Leslie, A. | title = Facial Attractiveness - Evolutionary, Cognitive, and Social Perspectives | publisher = [[Ablex]] | year = 2002 | isbn = 1-56750-636-4}}</ref><ref name="Jones2001">Jones, B. C., Little, A. C., Tiddeman, B. P., Burt, D. M., & Perrett, D. I. (2001). Facial symmetry and judgements of apparent health Support for a “‘ good genes ’” explanation of the attractiveness – symmetry relationship, 22, 417–429.</ref> 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>
===In literature===
Symmetry can be found in various forms in [[literature]], a simple example being the [[palindrome]] where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, as in the rise:fall pattern of ''[[Beowulf]]''.
==See also==
{{colbegin|2}}
*[[Burnside's lemma]]
*[[Chirality (mathematics)|Chirality]]
*[[Even and odd functions]]
*[[Fixed points of isometry groups in Euclidean space]] – center of symmetry
*[[Spacetime symmetries]]
*[[Spontaneous symmetry breaking]]
*[[Symmetry-breaking constraints]]
*[[Symmetric relation]]
*[[Polyiamond#Symmetries|Symmetries of polyiamonds]]
*[[Free polyomino|Symmetries of polyominoes]]
*[[Symmetry group]]
*[[T-symmetry|Time symmetry]]
*[[Wallpaper group]]
{{colend}}
==Notes==
{{notelist}}
==References==
{{reflist|26em}}
==
* ''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
==External links==
{{Wiktionary}}
{{Commons category|Symmetry}}
*[http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane]
*[http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
*[http://www.mi.sanu.ac.rs/~jablans/isis0.htm ISIS Symmetry]
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