Symmetry (from

File:Major and minor triads, triangles.png, Major and minor triads on the white piano keys are symmetrical to the D. (file)
poly 35 442 35 544 179 493 root of A minor triad
poly 479 462 446 493 479 526 513 492 third of A minor triad
poly 841 472 782 493 840 514 821 494 fifth of A minor triad
poly 926 442 875 460 906 493 873 525 926 545 fifth of A minor triad
poly 417 442 417 544 468 525 437 493 469 459
poly 502 472 522 493 502 514 560 493
poly 863 462 830 493 863 525 895 493
poly 1303 442 1160 493 1304 544
poly 280 406 264 413 282 419 275 413 fifth of E minor triad
poly 308 397 293 403 301 412 294 423 309 428 fifth of E minor triad
poly 844 397 844 428 886 413 root of E minor triad
poly 1240 404 1230 412 1239 422 1250 412 third of E minor triad
poly 289 404 279 413 288 422 300 413 third of G major triad
poly 689 398 646 413 689 429 fifth of G major triad
poly 1221 397 1222 429 1237 423 1228 414 1237 403 root of G major triad
poly 1249 406 1254 413 1249 418 1265 413 root of G major triad
poly 89 567 73 573 90 579 86 573 fifth of D minor triad
poly 117 558 102 563 111 572 102 583 118 589 fifth of D minor triad
poly 650 558 650 589 693 573 root of D minor triad
poly 1050 563 1040 574 1050 582 1061 574 third of D minor triad
poly 98 565 88 573 98 583 110 574 third of F major triad
poly 498 558 455 573 498 589 fifth of F major triad
poly 1031 557 1031 589 1047 583 1038 574 1046 563 root of F major triad
poly 1075 573 1059 580 1064 573 1058 567 root of F major triad
desc none
Symmetry is not restricted to the visual arts. Its role in the history of

Dutch: Symmetry Around a Point in the Plane

Symmetry

BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (''In Our Time'', Apr. 19, 2007) {{Patterns in nature Symmetry, Geometry Theoretical physics Artistic techniques Aesthetics

Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...

συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

; including translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

, reflectionReflection or reflexion may refer to:
Philosophy
* Self-reflection
Science
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal r ...

, rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

or scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energy ...

. Although these two meanings of "symmetry" 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
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

; as a ; through geometric transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s; through other kinds of functional transformations; and as an aspect of abstract object
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include Plant, plant ...

s, including theoretic models, language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...

, and music
Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated concepts , , and ...

.
This article describes symmetry from three perspectives: in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, including geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, the most familiar type of symmetry for many people; in science
Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

and nature
Nature, in the broadest sense, is the natural, physical, material world or universe
The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter an ...

; and in the arts, covering architecture
upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734.
Architecture (Latin ''archi ...

, art
Art is a diverse range of (products of) human activities
Humans (''Homo sapiens'') are the most populous and widespread species of primates, characterized by bipedality, opposable thumbs, hairlessness, and intelligence allowing the use o ...

and music
Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated concepts , , and ...

.
The opposite of symmetry is asymmetry
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in prec ...

, which refers to the absence or a violation of symmetry.
In mathematics

In geometry

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
250px, Figures with the axes of asymmetric.">asymmetry.html" ;"title="symmetry drawn in. The figure with no axes is asymmetry">asymmetric.
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to Re ...

(line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other.
*An object has rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...

if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.
*An object has translational symmetry
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

if it can be translated
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

(moving every point of the object by the same distance) without changing its overall shape.
*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
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rot ...

.
*An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractals
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

also exhibit a form of scale symmetry, where smaller portions of the fractal are similar in shape to larger portions.
*Other symmetries include glide reflection
In 2-dimensional geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relat ...

symmetry (a reflection followed by a translation) and rotoreflection
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

symmetry (a combination of a rotation and a reflection).
In logic

A 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''. 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 connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

s include ''and
And or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar)
In grammar
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study o ...

'' (∧, or &), '' or'' (∨, or , ) and ''if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

'' (↔), while the connective ''if'' (→) is not symmetric. Other symmetric logical connectives include '''' (not-and, or ⊼), ''xor
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, ...

'' (not-biconditional, or ⊻), and ''nor
Nor may refer to:
*Nor, a word used with "neither" in a correlative conjunction (e.g. "Neither the basketball team ''nor'' the football team is doing well.")
*Nor, a word used as a coordinating conjunction (e.g. "They do not gamble, ''nor'' do the ...

'' (not-or, or ⊽).
Other areas of mathematics

Generalizing from geometrical symmetry in the previous section, one can say that amathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...

is ''symmetric'' with respect to a given mathematical operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

, 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
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions
File:Développement limité du cosinus.svg, The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4.
In mathematics, even functions and odd functions are function ...

in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s in abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, symmetric matrices in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, and Galois group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s in Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

. In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

, symmetry also manifests as symmetric probability distribution
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...

s, and as skewness
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

—the asymmetry of distributions.
In science and nature

In physics

Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has become one of the most powerful tools oftheoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...

, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate 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
Noether's theorem or Noether's first theorem states that every differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...

(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 include continuous symmetries and discrete symmetries
In the modern world, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's origina ...

of spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

; internal symmetries
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some Transformation (function), transformation.
A family of particular transfo ...

of particles; and supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical physics, theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of super ...

of physical theories.
In biology

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 plane
In anatomy
Anatomy (Greek ''anatomē'', 'dissection') is the branch of biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molec ...

which 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. The 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.
Plants and sessile (attached) animals such as sea anemone
Sea anemones are the marine, predatory
Predation is a biological interaction
In ecology
Ecology (from el, οἶκος, "house" and el, -λογία, label=none, "study of") is the study of the relationships between living organisms ...

s often have radial or rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...

, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms
An echinoderm () is any member of the phylum Echinodermata (; ) of marine life, marine animals. The adults are recognizable by their (usually five-point) radial symmetry, and include starfish, sea urchins, sand dollars, and sea cucumbers, as we ...

, the group that includes starfish
Starfish or sea stars are star-shaped
In mathematics, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for ...

, sea urchin
Sea urchins () are spine (zoology), spiny, globular echinoderms in the class Echinoidea. About 950 species of sea urchin live on the seabed of every ocean and inhabit every depth zone — from the intertidal seashore down to . The spherical, ha ...

s, and sea lilies
Crinoids are marine animals that make up the Class (biology), class Crinoidea, one of the classes of the phylum Echinodermata, which also includes the starfish, brittle stars, sea urchins and sea cucumbers. Those crinoids which, in their adult f ...

.
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

Symmetry is important tochemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

because it undergirds essentially all ''specific'' interactions between molecules in nature (i.e., via the interaction of natural and human-made chiral
Chirality is a property of important in several branches of science. The word ''chirality'' is derived from the (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from its ; that is, i ...

molecules with inherently chiral biological systems). The control of the symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

of molecules produced in modern chemical synthesis
As a topic of chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they unde ...

contributes to the ability of scientists to offer therapeutic
A therapy or medical treatment (often abbreviated tx, Tx, or Tx) is the attempted remediation of a health
Health is a state of physical, mental and social well-being
Well-being, also known as ''wellness'', ''prudential value'' or ''quality ...

interventions with minimal side effects
In medicine, a side effect is an effect, whether therapeutic or adverse, that is secondary to the one intended; although the term is predominantly employed to describe adverse effects, it can also apply to beneficial, but unintended, consequences ...

. A rigorous understanding of symmetry explains fundamental observations in quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of chemistry focused on the application of quantum mechanics to chemical systems. Understanding electronic structure and molecular dynamics using the Schrödinger equations a ...

, and in the applied areas of spectroscopy
Spectroscopy is the study of the interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ...

and crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "cold drop, frozen drop" ...

. The theory and application of symmetry to these areas of physical science
Physical science is a branch of natural science
Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observa ...

draws heavily on the mathematical area of group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

.
In psychology and neuroscience

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
Ernst Waldfried Josef Wenzel Mach (; ; 18 February 1838 – 19 February 1916) was an Austrian physicist and philosopher
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , t ...

made 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 the Gestalt
''Gestalt'', a German word for form or shape, may refer to:
* Holism, the idea that natural systems and their properties should be viewed as wholes, not as loose collections of parts
Psychology
* Gestalt psychology (also known as "Gestalt theory" ...

tradition suggested that bilateral symmetry was one of the key factors in perceptual grouping. This is known as the 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. 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
Reciprocity may refer to:
Law and trade
* Reciprocity (Canadian politics), free trade with the United States of America
** Reciprocal trade agreement, entered into in order to reduce (or eliminate) tariffs, quotas and other trade restrictions on ...

, empathy
Empathy is the capacity to understand or feel what another person is experiencing from within their frame of reference, that is, the capacity to place oneself in another's position. Definitions of empathy encompass a broad range of emotional s ...

, sympathy
Sympathy is the perception, understanding, and reaction to the distress or need of another life form. According to David Hume, this sympathetic concern is driven by a switch in viewpoint from a personal perspective to the perspective of another g ...

, apology
Apology, apologize/apologise, apologist, apologetics, or apologetic may refer to:
Common uses
* Apology (act)
An apology is an expression of regret or remorse for actions, while apologizing is the act of expressing regret or remorse. In inform ...

, dialogue
Dialogue (sometimes spelled dialog in American English
American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native ...

, respect, justice
Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and perspectives, ...

, and revenge
Revenge is committing a harmful action against a person or group in response to a grievance
A grievance () is a wrong or hardship suffered, real or supposed, which forms legitimate grounds of complaint. In the past, the word meant the infl ...

.
Reflective equilibrium
Reflective equilibrium is a state of balance or coherence among a set of beliefs arrived at by a process of deliberative mutual adjustment among general principles and particular judgments. Although he did not use the term, philosopher Nelson Go ...

is the balance that may be attained through deliberative mutual adjustment among general principles and specific judgment
Judgement (or US spelling judgment) is also known as ''adjudication
Adjudication is the legal process by which an arbitration, arbiter or judge reviews evidence (law), evidence and argumentation, including legal reasoning set forth by opposing ...

s.
Symmetrical interactions send the moral
A moral (from Latin ''morālis'') is a message that is conveyed or a lesson to be learned from a narrative, story or wikt:event, event. The moral may be left to the hearer, reader, or viewer to determine for themselves, or may be explicitly enca ...

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
The Golden Rule is the principle of treating others as one wants to be treated. It is a maxim that is found in most religion
Religion is a social system, social-cultural system of designated religious behaviour, behaviors and practices, ...

, are based on symmetry, whereas power relationships are based on asymmetry. Symmetrical relationships can to some degree be maintained by simple (game theory
Game theory is the study of mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

) strategies seen in symmetric gamesIn game theory
Game theory is the study of mathematical models of strategic interactions among Rational agent, rational agents.Roger B. Myerson, Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 C ...

such as tit for tat
In Western business cultures, a handshake when meeting someone is a signal of initial cooperation.
Tit for tat is an English saying meaning "equivalent retaliation". It developed from "tip for tap", first recorded in 1558.
It is also a highly ...

.
In the arts

It exists a list of journals and newsletters known to deal, at least for a part, with symmetry and arts.In architecture

Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothiccathedral
A cathedral is a church
Church may refer to:
Religion
* Church (building)
A church building, church house, or simply church, is a building used for Christian worship services and other Christian religious activities. The term is used ...

s and The White House
The White House is the official residence
An official residence is the House, residence of nation's head of state, head of government, governor, Clergy, religious leader, leaders of international organizations, or other senior figure. I ...

, through the layout of the individual floor plan
In architecture
File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gal ...

s, and down to the design of individual building elements such as . Islam
Islam (; ar, اَلْإِسْلَامُ, al-’Islām, "submission o God
Oh God may refer to:
* An exclamation; similar to "oh no", "oh yes", "oh my", "aw goodness", "ah gosh", "ah gawd"; see interjection
An interjection is a word or ex ...

ic buildings such as the Taj Mahal
The Taj Mahal (; , ), is an ivory-white marble mausoleum on the right bank of the river Yamuna in the Indian city of Agra. It was commissioned in 1632 by the Mughal Empire, Mughal emperor Shah Jahan () to house the tomb of his favourite wi ...

and the make elaborate use of symmetry both in their structure and in their ornamentation. Moorish buildings like the Alhambra
The Alhambra (, ; ar, الْحَمْرَاء, Al-Ḥamrāʾ, , ) is a palace and fortress complex located in Granada
Granada ( , ,, DIN 31635, DIN: ; grc, Ἐλιβύργη, Elibýrgē; la, Illiberis or . ) is the capital city of the provi ...

are 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
Modern architecture, or modernist architecture, was an architectural movement or architectural style
An architectural style is a set of characteristics and features that make a building or other structure notable or historically identifiable. ...

, starting with International style, relies instead on "wings and balance of masses".
In pottery and metal vessels

Since the earliest uses of pottery wheels 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 ancientChinese
Chinese can refer to:
* Something related to China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's most populous country, with a populat ...

, 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

A long tradition of the use of symmetry incarpet
A carpet is a textile
A textile is a flexible material made by creating an interlocking bundle of yarn
Yarn is a long continuous length of interlocked fibres, suitable for use in the production of textiles, sewing, crocheting, kni ...

and rug patterns spans a variety of cultures. American Navajo
The Navajo (; British English: Navaho; nv, Diné or ') are a of the .
At more than 399,494 enrolled tribal members , the is the largest federally recognized tribe in the U.S. (the being the second largest); the Navajo Nation has the larges ...

Indians used bold diagonals and rectangular motifs. Many Oriental rugs
An oriental rug is a heavy textile made for a wide variety of utilitarian and symbolic purposes and produced in "Orient, Oriental countries" for home use, local sale, and export.
Oriental carpets can be knotted-pile carpet, pile woven or Kilim, ...

have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

—that is, motifs that are reflected across both the horizontal and vertical axes (see ).
In music

music
Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated concepts , , and ...

touches many aspects of the creation and perception of music.
Musical form

Symmetry has been used as aformal
Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for forma ...

constraint by many composers, such as the (ABCBA) used by Steve Reich
Stephen Michael Reich ( ; born October 3, 1936) is an American composer known for his contribution to the development of minimal music
Minimal music (also called minimalism)"Minimalism in music has been defined as an aesthetic, a style, an ...

, Béla Bartók
Béla Viktor János Bartók (; hu, Bartók Béla, ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt ...

, and James Tenney
James Tenney (August 10, 1934 – August 24, 2006) was an American composer
A composer (Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoke ...

. In classical music, Bach used the symmetry concepts of permutation and invariance.
Pitch structures

Symmetry is also an important consideration in the formation of scale (music), scales 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 scales 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
Béla Viktor János Bartók (; hu, Bartók Béla, ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt ...

, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to musical key, keys or non-tonality, tonal tonal Tonic (music), centers. George Perle 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:"
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).
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 String Quartet (Berg), ''Quartet'', Op. 3 (1910).
Equivalency

Tone rows 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 quilts

As quilts 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.In other arts and crafts

Symmetries appear in the design of objects of all kinds. Examples include beadwork, furniture, sand paintings, knotwork, 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. 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 aesthetics

The relationship of symmetry to aesthetics is complex. Humans find bilateral symmetry in faces physically attractive; it indicates health and genetic fitness.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. 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.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''.See also

*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 groupNotes

References

Further reading

* ''The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry'', Mario Livio, Souvenir Press 2006,External links

Dutch: Symmetry Around a Point in the Plane

Symmetry

BBC Radio 4 discussion with Fay Dowker, Marcus du Sautoy & Ian Stewart (''In Our Time'', Apr. 19, 2007) {{Patterns in nature Symmetry, Geometry Theoretical physics Artistic techniques Aesthetics