Editing Symmetry

{{Short description|Mathematical invariance under transformations}}
{{About|the broad concept}}
[[File:Asymmetric (PSF).svg|thumb|upright=1.25|Symmetry (left) and [[asymmetry]] (right)]]
[[File:Sphere symmetry group o.svg|thumb|upright=0.8|A spherical [[symmetry group]] with [[octahedral symmetry]]. The yellow region shows the [[fundamental domain]].]]
[[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]].]]
{{General geometry}}

'''Symmetry''' ({{etymology|grc|''{{Wikt-lang|grc|συμμετρία}}'' ({{grc-transl|συμμετρία}})|agreement in dimensions, due proportion, arrangement}})<ref>{{OEtymD|symmetry}}</ref> in everyday life refers to a sense of harmonious and beautiful proportion and balance.<ref>{{cite book |last=Zee|first=A. |title=Fearful Symmetry |publisher=[[Princeton University Press]] |location=[[Princeton, New Jersey]] |year=2007 |isbn=978-0-691-13482-6}}</ref><ref>{{cite book |title=Symmetry and the Beautiful Universe |last1=Hill|first1=C. T. |author1-link=Christopher T. Hill |last2=Lederman|first2=L. M. |author2-link=Leon M. Lederman |publisher=[[Prometheus Books]] |year=2005}}</ref>{{efn|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.}} In [[mathematics]], the term has a more precise definition and is usually used to refer to an object that is [[Invariant (mathematics)|invariant]] under some [[Transformation (function)|transformations]], such as [[Translation (geometry)|translation]], [[Reflection (mathematics)|reflection]], [[Rotation (mathematics)|rotation]], or [[Scaling (geometry)|scaling]]. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.

Mathematical symmetry may be observed with respect to the passage of [[time]]; as a [[space|spatial relationship]]; through [[geometric transformation]]s; through other kinds of functional transformations; and as an aspect of [[abstract object]]s, including [[scientific model|theoretic models]], [[language]], and [[music]].<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]], which refers to the absence of symmetry.

==In mathematics==

===In geometry===
{{main|Symmetry (geometry)}}
[[File:The armoured triskelion on the flag of the Isle of Man.svg|thumb|upright=0.8|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 (book)|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]] (line or mirror symmetry) if there is a line (or in [[three-dimensional space|3D]] a plane) going through it which divides it into two pieces that are mirror images of each other.<ref>{{cite book |title=Symmetry |last=Weyl |first=Hermann |author-link=Hermann Weyl |year=1982 |orig-year=1952 |publisher=Princeton University Press |location=Princeton | isbn=0-691-02374-3 |ref=Weyl 1982}}</ref>
*An object has [[rotational symmetry]] if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.<ref>{{cite book | author=Singer, David A. | year=1998 | title=Geometry: Plane and Fancy | url=https://archive.org/details/geometryplanefan0000sing | url-access=registration | publisher=Springer Science & Business Media}}</ref>
*An object has [[translational symmetry]] if it can be [[translation (geometry)|translated]] (moving every point of the object by the same distance) 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 smaller portions of the fractal are [[similarity (geometry)|similar]] in shape to larger 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 (a reflection followed by a translation) and [[improper rotation|rotoreflection]] symmetry (a combination of a rotation and a reflection<ref>{{Cite web|url=https://encyclopedia2.thefreedictionary.com/rotoreflection+axis|title=Rotoreflection Axis|website=TheFreeDictionary.com|access-date=2019-11-12}}</ref>).

===In logic===

A [[binary relation|dyadic relation]] ''R'' = ''S'' × ''S'' is symmetric if for all elements ''a'', ''b'' in ''S'', whenever it is true that ''Rab'', it is also 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, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.

In propositional logic, symmetric binary [[logical connective]]s include ''[[logical conjunction|and]]'' (∧, or &), ''[[logical disjunction|or]]'' (∨, or |) and ''[[if and only if]]'' (↔), while the connective ''if'' (→) is not symmetric.<ref>{{Cite web|url=https://cs.uwaterloo.ca/~a23gao/cs245_f19/slides/lec02_prop_syntax_nosol.pdf|title=Propositional Logic: Introduction and Syntax|last=Gao|first=Alice|date=2019|website=University of Waterloo — School of Computer Science|access-date=2019-11-12}}</ref> Other symmetric logical connectives include ''[[logical nand|nand]]'' (not-and, or ⊼), ''[[xor]]'' (not-biconditional, or ⊻), and ''[[logical nor|nor]]'' (not-or, or ⊽).

===Other areas of mathematics===
{{main|Symmetry in mathematics}}
Generalizing from geometrical symmetry in the previous section, one can 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]], [[symmetric group]]s in [[abstract algebra]], [[symmetric matrix|symmetric matrices]] in [[linear algebra]], and [[Galois group]]s in [[Galois theory]]. In [[statistics]], symmetry also manifests as [[symmetric probability distribution]]s, and as [[skewness]]—the asymmetry of distributions.<ref>{{cite journal | author = Petitjean, M. | title = Chirality and Symmetry Measures: A Transdisciplinary Review | journal = Entropy | year = 2003 | volume = 5 | issue = 3 | pages=271–312 (see section 2.9) | doi = 10.3390/e5030271| bibcode = 2003Entrp...5..271P | doi-access = free }}</ref>

==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 |page = 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 |bibcode = 1972Sci...177..393A | s2cid=34548824 }}</ref> See [[Noether's theorem]] (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language);<ref name=Noether>{{Cite book | last = Kosmann-Schwarzbach | first = Yvette | author-link = 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|author-link=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 |bibcode = 1939AnMat..40..149W |jstor=1968551|s2cid=121773411 }}</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.

===In biology===
{{Further|symmetry in biology|facial symmetry}}
[[File:Chance and a Half, Posing.jpg|thumb|upright=0.8|Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.]]
In biology, the notion of symmetry is mostly used explicitly to describe body shapes. [[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 |access-date=29 May 2013 |url-status=dead |archive-url=https://web.archive.org/web/20080118213208/http://www.accessscience.com/abstract.aspx?id=802620&referURL=http%3A%2F%2Fwww.accessscience.com%2Fcontent.aspx%3Fid%3D802620 |archive-date=18 January 2008 }}</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 | access-date=October 25, 2012 | author1=Hickman, Cleveland P. | author2=Roberts, Larry S. | author3=Larson, Allan | page=139 | archive-date=May 17, 2016 | archive-url=http://arquivo.pt/wayback/20160517212058/http://biocongroup.eu/DA/Calendario_files/Bilateria.pdf | url-status=dead }}</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 biology, the notion of symmetry is also used as in physics, that is to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics.<ref>{{cite book |url=https://www.springer.com/la/book/9783642359378 |title=Perspectives on Organisms: Biological time, Symmetries and Singularities |last1=Longo |first1=Giuseppe |last2=Montévil |first2=Maël |date=2016 |publisher=Springer  |isbn=978-3-662-51229-6 |language=en}}</ref><ref>{{cite journal |last1=Montévil |first1=Maël |last2=Mossio |first2=Matteo |last3=Pocheville |first3=Arnaud |last4=Longo |first4=Giuseppe |date=2016 |title=Theoretical principles for biology: Variation |url=https://www.academia.edu/27942089 |journal=Progress in Biophysics and Molecular Biology |series=From the Century of the Genome to the Century of the Organism: New Theoretical Approaches |volume=122 |issue=1 |pages=36–50 |doi=10.1016/j.pbiomolbio.2016.08.005|pmid=27530930 |s2cid=3671068 }}</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 |author1=Lowe, John P |author2=Peterson, Kirk | title=Quantum Chemistry | publisher=Academic Press| edition=Third | year=2005 | isbn=0-12-457551-X}}</ref>

===In psychology and neuroscience===
{{Further|Visual perception}}

For a human observer, some symmetry types are more salient than others, in particular the most salient is a reflection with a vertical axis, like that present in the human face. [[Ernst Mach]] made this observation in his book "The analysis of sensations" (1897),<ref>{{cite book |title = Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries | first1 = Ernst |last1 = Mach | publisher = Open Court Publishing House |year = 1897}}</ref> and this implies that perception of symmetry is not a general response to all types of regularities. Both behavioural and neurophysiological studies have confirmed the special sensitivity to reflection symmetry in humans and also in other animals.<ref>{{cite journal|author1=Wagemans, J.|title=Characteristics and models of human symmetry detection|journal=[[Trends in Cognitive Sciences]]|volume=1|issue=9|pages=346–352| year=1997|doi= 10.1016/S1364-6613(97)01105-4|pmid=21223945|s2cid=2143353|url=https://lirias.kuleuven.be/handle/123456789/207060}}</ref> Early studies within the [[Gestalt psychology|Gestalt]] tradition suggested that bilateral symmetry was one of the key factors in perceptual [[Principles of grouping|grouping]]. This is known as the [[Gestalt psychology#Law of Symmetry|Law of Symmetry]]. The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry is faster when this is a property of a single object.<ref>{{cite journal | author1=Bertamini, M.| title=Sensitivity to reflection and translation is modulated by objectness | journal=[[Perception (journal)|Perception]]| volume=39|pages=27–40| year=2010| issue=1 | doi=10.1068/p6393| pmid=20301844 | s2cid=22451173 }}</ref> Studies of human perception and psychophysics have shown that detection of symmetry is fast, efficient and robust to perturbations. For example, symmetry can be detected with presentations between 100 and 150 milliseconds.<ref>{{cite journal|author1=Barlow, H.B.|author2=Reeves, B.C.|title=The versatility and absolute efficiency of detecting mirror symmetry in random dot displays|journal=[[Vision Research]]|volume=19|issue=7|pages=783–793|year=1979|doi= 10.1016/0042-6989(79)90154-8|pmid=483597|s2cid=41530752}}</ref>

More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al.<ref>{{cite journal |author1=Sasaki, Y.|author2=Vanduffel, W.|author3=Knutsen, T.|author4=Tyler, C.W.|author5=Tootell, R.|title=Symmetry activates extrastriate visual cortex in human and nonhuman primates |journal=[[Proceedings of the National Academy of Sciences of the USA]]|volume=102|issue=8|pages=3159–3163|year=2005|doi= 10.1073/pnas.0500319102|pmid=15710884|pmc=549500|bibcode=2005PNAS..102.3159S|doi-access=free}}</ref> used functional magnetic resonance imaging (fMRI) to compare responses for patterns with symmetrical or random dots. A strong activity was present in extrastriate regions of the occipital cortex but not in the primary visual cortex. The extrastriate regions included V3A, V4, V7, and the lateral occipital complex (LOC). Electrophysiological studies have found a late posterior negativity that originates from the same areas.<ref>{{cite journal |author1=Makin, A.D.J. |author2=Rampone, G. |author3= Pecchinenda, A. |author4= Bertamini, M. |title= Electrophysiological responses to visuospatial regularity |journal=[[Psychophysiology]]| volume=50| pages= 1045–1055|year=2013|issue=10 |doi=10.1111/psyp.12082|pmid=23941638 }}</ref> In general, a large part of the visual system seems to be involved in processing visual symmetry, and these areas involve similar networks to those responsible for detecting and recognising objects.<ref>{{cite journal |author1=Bertamini, M.|author2=Silvanto, J. |author3=Norcia, A.M. |author4=Makin, A.D.J. |author5= Wagemans, J. |title=The neural basis of visual symmetry and its role in middle and high-level visual processing |journal=[[Annals of the New York Academy of Sciences]]|volume=132|pages=280–293|year=2018|issue=1 |doi=10.1111/nyas.13667|pmid=29604083 |bibcode=2018NYASA1426..111B |doi-access=free|hdl=11577/3289328 |hdl-access=free }}</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]], [[Remorse|apology]], [[dialogue]], 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>{{cite SEP |url-id=reflective-equilibrium |title=Reflective Equilibrium |last=Daniels |first=Norman |author-link=Norman Daniels |date=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|access-date=28 September 2015|date=2008}}</ref>

==In the arts==
{{Further|Mathematics and art}}

There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.<ref>{{cite journal | last1 = Bouissou, C. | last2 = Petitjean, M. | title = Asymmetric Exchanges | journal = Journal of Interdisciplinary Methodologies and Issues in Science | year = 2018 | volume = 4 | pages = 1–18 | url = https://hal.archives-ouvertes.fr/hal-01782438v2/document | doi = 10.18713/JIMIS-230718-4-1 | doi-access = free}} (see  appendix 1)</ref>

===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=https://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 news |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 |newspaper=New York Times |access-date=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|right|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] {{Webarchive|url=https://web.archive.org/web/20031211185121/http://chinavoc.com/arts/handicraft/bronze.htm |date=2003-12-11 }}. Chinavoc (2007-11-19). Retrieved on 2013-04-16.</ref>

===In carpets and rugs===
[[File:Farsh1.jpg|thumb|upright=1.5|Persian rug with rectangular symmetry]]

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 have typically the symmetries of a [[rectangle]]—that is, [[Motif (visual arts)|motifs]] that are reflected across both the horizontal and vertical axes (see {{slink|Klein four-group|Geometry}}).<ref>[https://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 quilts===
[[File:kitchen kaleid.svg|thumb|upright=0.65|Kitchen [[kaleidoscope]] [[quilt]] 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] {{Webarchive|url=https://web.archive.org/web/20031231031119/http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm |date=2003-12-31 }}. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.</ref>

===In other arts and crafts===
{{Further|Islamic geometric patterns}}

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 such as in [[Islamic geometric patterns|Islamic geometric decoration]], [[batik]], [[ikat]], carpet-making, and many kinds of [[textile]] and [[embroidery]] patterns.<ref>{{cite book |last1=Cucker |first1=Felipe |author1-link=Felipe Cucker|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>

Symmetry is also used in designing logos.<ref>{{cite web|title=How to Design a Perfect Logo with Grid and Symmetry|url=https://www.designmantic.com/how-to/how-to-design-a-perfect-logo}}</ref> By creating a logo on a grid and using the theory of symmetry, designers can organize their work, create a symmetric or asymmetrical design, determine the space between letters, determine how much negative space is required in the design, and how to accentuate parts of the logo to make it stand out.

===In music===
[[File:Major and minor triads, circles, dozenal.png|thumb|[[Major and minor]] triads on the white piano keys are symmetrical to the D.]]

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, [[Johann Sebastian Bach]] used the symmetry concepts of permutation and invariance.<ref>see ("Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] {{Webarchive|url=https://web.archive.org/web/20050913002304/http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf |date=2005-09-13 }} or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave] {{Webarchive|url=https://web.archive.org/web/20051026015256/http://jan.ucc.nau.edu/~tas3/wtc/ii21.html |date=2005-10-26 }})</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.<ref name=Perle1992>{{Cite journal |title=Symmetry, the twelve-tone scale, and tonality |first=George |last=Perle |author-link=George Perle |journal=Contemporary Music Review |volume=6 |issue=2 |year=1992 |pages=81–96 |doi=10.1080/07494469200640151}}</ref> George Perle explains that "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:"<ref name=Perle1992/>

{|
|-
|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).<ref name=Perle1992/>

{|
|rowspan=3|+
|2
|
|3
|
|'''4'''
|
|5
|
|6
|
|7
|
|8
|-
|2
|
|1
| 
|'''0'''
|
|11
|
|10
|
|9
|
|8
|-
|4
|
|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.<ref name=Perle1992/><ref name="Perle1990">{{cite book |author-link=George Perle |author=Perle, George |year=1990 |title=The Listening Composer |url=https://archive.org/details/listeningcompose00perl |url-access=limited |page=[https://archive.org/details/listeningcompose00perl/page/n31 21] |publisher=University of California Press |isbn=978-0-520-06991-6}}</ref>

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's [[String Quartet (Berg)|''Quartet'', Op. 3]] (1910).<ref name="Perle1990"/>

====Equivalency====

[[Tone row]]s or [[pitch class]] [[Set theory (music)|sets]] which are [[Invariant (music)|invariant]] under [[Permutation (music)|retrograde]] are horizontally symmetrical, under [[Melodic inversion|inversion]] vertically. See also [[Asymmetric rhythm]].

===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">{{cite journal |last1=Grammer |first1=K. |last2=Thornhill |first2=R. |date=1994 |title=Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness |journal=Journal of Comparative Psychology |location=Washington, D.C. |volume=108 |issue=3 |pages=233–42 |doi=10.1037/0735-7036.108.3.233|pmid=7924253 |s2cid=1205083 }}</ref> it indicates health and genetic fitness.<ref>{{cite book |last1=Rhodes |first1=Gillian |last2=Zebrowitz |first2=Leslie A. |author2-link=Leslie Zebrowitz |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. Rudolf Arnheim suggested that people prefer shapes that have some symmetry, and enough complexity to make them interesting.<ref>{{cite book |last=Arnheim |first=Rudolf |title=Visual Thinking |url=https://archive.org/details/visualthinking00rudo |url-access=registration |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, such as the rise and fall pattern of ''[[Beowulf]]''.<ref>{{cite web | url=http://trace.tennessee.edu/cgi/viewcontent.cgi?article=1925&context=utk_gradthes | title=Symmetrical Aesthetics of Beowulf | publisher=University of Tennessee, Knoxville | year=2009 |author1=Jenny Lea Bowman}}</ref>

==See also==
{{Portal|Philosophy|Astronomy|Architecture|Mathematics|Physics|Star|Clothing}}
{{Div col|colwidth=18em}}
*[[Automorphism]]
*[[Burnside's lemma]]
*[[Chirality (mathematics)|Chirality]]
*[[Even and odd functions]]
*[[Fixed points of isometry groups in Euclidean space]] – center of symmetry
*[[Isotropy]]
*[[Palindrome]]
*[[Spacetime symmetries]]
*[[Spontaneous symmetry breaking]]
*[[Symmetry-breaking constraints]]
*[[Symmetric relation]]
*[[Polyiamond#Symmetries|Symmetries of polyiamonds]]
*[[Free polyomino|Symmetries of polyominoes]]
*[[Symmetry group]]
*[[Wallpaper group]]
{{div col end}}

== Explanatory notes ==
{{Notelist}}

==References==
{{Reflist|26em}}

==Further reading==
* ''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}}
{{Wikiquote}}
* [https://isa.symmetry.hu International Symmetry Association (ISA)]
* [http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane] {{Webarchive|url=https://web.archive.org/web/20040102035006/http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM |date=2004-01-02 }}
* [http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
* [http://www.mi.sanu.ac.rs/~jablans/isis0.htm ISIS Symmetry] {{Webarchive|url=https://web.archive.org/web/20090922133038/http://www.mi.sanu.ac.rs/~jablans/isis0.htm |date=2009-09-22 }}
* [https://www.bbc.co.uk/programmes/b00776v8 Symmetry], BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (''In Our Time'', Apr. 19, 2007)

{{Mathematics and art}}
{{Patterns in nature}}
{{Authority control}}

[[Category:Symmetry| ]]
[[Category:Aesthetics]]
[[Category:Artistic techniques]]
[[Category:Geometry]]
[[Category:Theoretical physics]]