Symmetry

This article is about the broad concept. For other uses, seeSymmetry (disambiguation).

Symmetry (left) andasymmetry(right)

A sphericalsymmetry groupwithoctahedral symmetry.The yellow region shows thefundamental domain.

Afractal-like shape that hasreflectional symmetry,rotational symmetryandself-similarity,three forms of symmetry. This shape is obtained by afinite subdivision rule.

Symmetry(fromAncient Greekσυμμετρία(summetría)'agreement in dimensions, due proportion, arrangement')in everyday life refers to a sense of harmonious and beautiful proportion and balance.Inmathematics,the term has a more precise definition and is usually used to refer to an object that isinvariantunder sometransformations,such astranslation,reflection,rotation,orscaling.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 oftime;as aspatial relationship;throughgeometric transformations;through other kinds of functional transformations; and as an aspect ofabstract objects,includingtheoretic models,language,andmusic.

This article describes symmetry from three perspectives: inmathematics,includinggeometry,the most familiar type of symmetry for many people; inscienceandnature;and in the arts, coveringarchitecture,art,and music.

The opposite of symmetry isasymmetry,which refers to the absence of symmetry.

In mathematics

In geometry

Main article:Symmetry (geometry)

Thetriskelionhas 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.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 hasreflectional symmetry(line or mirror symmetry) if there is a line (or in3Da plane) going through it which divides it into two pieces that are mirror images of each other.
• An object hasrotational symmetryif the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.
• An object hastranslational symmetryif it can betranslated(moving every point of the object by the same distance) without changing its overall shape.
• An object hashelical symmetryif it can be simultaneously translated and rotated in three-dimensional space along a line known as ascrew axis.
• An object hasscale symmetryif it does not change shape when it is expanded or contracted.Fractalsalso exhibit a form of scale symmetry, where smaller portions of the fractal aresimilarin shape to larger portions.
• Other symmetries includeglide reflectionsymmetry (a reflection followed by a translation) androtoreflectionsymmetry (a combination of a rotation and a reflection).

In logic

Adyadic relationR=S×Sis symmetric if for all elementsa,binS,whenever it is true thatRab,it is also true thatRba.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 binarylogical connectivesincludeand(∧, or &),or(∨, or |) andif and only if(↔), while the connectiveif(→) is not symmetric.Other symmetric logical connectives includenand(not-and, or ⊼),xor(not-biconditional, or ⊻), andnor(not-or, or ⊽).

Other areas of mathematics

Main article:Symmetry in mathematics

Generalizing from geometrical symmetry in the previous section, one can say that amathematical objectissymmetricwith respect to a givenmathematical 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 agroup.

In general, every kind of structure in mathematics will have its own kind of symmetry. Examples includeeven and odd functionsincalculus,symmetric groupsinabstract algebra,symmetric matricesinlinear algebra,andGalois groupsinGalois theory.Instatistics,symmetry also manifests assymmetric probability distributions,and asskewness—the asymmetry of distributions.

In science and nature

Further information:Patterns in nature

In physics

Main article:Symmetry in physics

Symmetry in physics has been generalized to meaninvariance—that is, lack of change—under any kind of transformation, for examplearbitrary coordinate transformations.This concept has become one of the most powerful tools oftheoretical physics,as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureatePW Andersonto write in his widely read 1972 articleMore is Differentthat "it is only slightly overstating the case to say that physics is the study of symmetry."SeeNoether'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);and also,Wigner's classification,which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

Important symmetries in physics includecontinuous symmetriesanddiscrete symmetriesofspacetime;internal symmetriesof particles; andsupersymmetryof physical theories.

In biology

Further information:symmetry in biologyandfacial symmetry

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.Bilateral animals,including humans, are more or less symmetric with respect to thesagittal planewhich divides the body into left and right halves.Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. Thehead becomes specializedwith 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.

Plants and sessile (attached) animals such assea anemonesoften have radial orrotational symmetry,which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in theechinoderms,the group that includesstarfish,sea urchins,andsea lilies.

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.

In chemistry

Main article:Molecular symmetry

Symmetry is important tochemistrybecause it undergirds essentially allspecificinteractions between molecules in nature (i.e., via the interaction of natural and human-madechiralmolecules with inherently chiral biological systems). The control of thesymmetryof molecules produced in modernchemical synthesiscontributes to the ability of scientists to offertherapeuticinterventions with minimalside effects.A rigorous understanding of symmetry explains fundamental observations inquantum chemistry,and in the applied areas ofspectroscopyandcrystallography.The theory and application of symmetry to these areas ofphysical sciencedraws heavily on the mathematical area ofgroup theory.

In psychology and neuroscience

Further information: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 Machmade this observation in his book "The analysis of sensations" (1897),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.Early studies within theGestalttradition suggested that bilateral symmetry was one of the key factors in perceptualgrouping.This is known as theLaw 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.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.

More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al.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.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.

In social interactions

People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments ofreciprocity,empathy,sympathy,apology,dialogue,respect,justice,andrevenge. Reflective equilibriumis the balance that may be attained through deliberative mutual adjustment among general principles and specificjudgments. Symmetrical interactions send themoralmessage "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 theGolden Rule,are based on symmetry, whereas power relationships are based on asymmetry.Symmetrical relationships can to some degree be maintained by simple (game theory) strategies seen insymmetric gamessuch astit for tat.

In the arts

Further information:Mathematics and art

There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.

In architecture

Further information:Mathematics and architecture

Seen from the side, theTaj Mahalhas 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 GothiccathedralsandThe White House,through the layout of the individualfloor plans,and down to the design of individual building elements such astile mosaics.Islamicbuildings such as theTaj Mahaland theLotfollah mosquemake elaborate use of symmetry both in their structure and in their ornamentation.Moorish buildings like theAlhambraare ornamented with complex patterns made using translational and reflection symmetries as well as rotations.

It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";Modernist architecture,starting withInternational style,relies instead on "wings and balance of masses".

In pottery and metal vessels

Clay pots thrown on apottery wheelacquire rotational symmetry.

Since the earliest uses ofpottery wheelsto 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 ancientChinese,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.

In carpets and rugs

Persian rug with rectangular symmetry

A long tradition of the use of symmetry incarpetand rug patterns spans a variety of cultures. AmericanNavajoIndians used bold diagonals and rectangular motifs. ManyOriental rugshave intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of arectangle—that is,motifsthat are reflected across both the horizontal and vertical axes (seeKlein four-group § Geometry).

In quilts

Kitchenkaleidoscopequiltblock

Asquiltsare 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.

In other arts and crafts

Further information:Islamic geometric patterns

Symmetries appear in the design of objects of all kinds. Examples includebeadwork,furniture,sand paintings,knotwork,masks,andmusical instruments.Symmetries are central to the art ofM.C. Escherand the many applications oftessellationin art and craft forms such aswallpaper,ceramic tilework such as inIslamic geometric decoration,batik,ikat,carpet-making, and many kinds oftextileandembroiderypatterns.

Symmetry is also used in designing logos.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

Major and minortriads on the white piano keys are symmetrical to the D.

Symmetry is not restricted to the visual arts. Its role in the history ofmusictouches many aspects of the creation and perception of music.

Musical form

Symmetry has been used as aformalconstraint by many composers, such as thearch (swell) form(ABCBA) used bySteve Reich,Béla Bartók,andJames Tenney.In classical music,Johann Sebastian Bachused the symmetry concepts of permutation and invariance.

Pitch structures

Symmetry is also an important consideration in the formation ofscalesandchords,traditional ortonalmusic being made up of non-symmetrical groups ofpitches,such as thediatonic scaleor themajor chord.Symmetrical scalesor chords, such as thewhole tone scale,augmented chord,or diminishedseventh chord(diminished-diminished seventh), are said to lack direction or a sense of forward motion, areambiguousas to thekeyor tonal center, and have a less specificdiatonic functionality.However, composers such asAlban Berg,Béla Bartók,andGeorge Perlehave used axes of symmetry and/orinterval cyclesin an analogous way tokeysor non-tonaltonalcenters.George Perle explains that "C–E, D–F♯, [and] Eb–G, are different instances of the sameinterval…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).

+	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 areenharmonicwith the cycle of fourths) will produce the diatonic major scale. Cyclic tonalprogressionsin the works ofRomanticcomposers such asGustav MahlerandRichard Wagnerform 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'sQuartet,Op. 3(1910).

Equivalency

Tone rowsorpitch classsetswhich areinvariantunderretrogradeare horizontally symmetrical, underinversionvertically. See alsoAsymmetric rhythm.

In aesthetics

Main article:Symmetry (physical attractiveness)

The relationship of symmetry toaestheticsis complex. Humans findbilateral symmetryin faces physically attractive;it indicates health and genetic fitness.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.

In literature

Symmetry can be found in various forms inliterature,a simple example being thepalindromewhere a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern ofBeowulf.

See also

• Automorphism
• Burnside's lemma
• 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
• Symmetries of polyiamonds
• Symmetries of polyominoes
• Symmetry group
• Wallpaper group

Further reading

• The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry,Mario Livio,Souvenir Press,2006,ISBN0-285-63743-6.

External links

• International Symmetry Association (ISA)
• Dutch: Symmetry Around a Point in the PlaneArchived2004-01-02 at theWayback Machine
• Chapman: Aesthetics of Symmetry
• ISIS SymmetryArchived2009-09-22 at theWayback Machine
• Symmetry,BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (In Our Time,Apr. 19, 2007)