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Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, "symmetry" has a more precise definition, and is usually used to refer to an object that is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under some transformations; including translation, reflection,
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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, energ ...
. 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; as a spatial relationship; through geometric transformations; 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 plants, hum ...
s, including theoretic models, language, and music. This article describes symmetry from three perspectives: in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 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 pre ...
, 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 has
reflectional symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
(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 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 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 In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symme ...
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 and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...
. *An object has
scale symmetry In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical te ...
if it does not change shape when it is expanded or contracted. Fractals 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, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
symmetry (a reflection followed by a translation) and rotoreflection symmetry (a combination of a rotation and a reflection).


In logic

A
dyadic relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
''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 binary logical connectives include ''
and or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boole ...
'' (∧, or &), '' or'' (∨, or , ) and '' if and only if'' (↔), while the connective ''if'' (→) is not symmetric. Other symmetric logical connectives include '' nand'' (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'' (not-or, or ⊽).


Other areas of mathematics

Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is ''symmetric'' with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory. In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, symmetry also manifests as
symmetric probability distribution In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) ...
s, and as skewness—the asymmetry of distributions.


In science and nature


In physics

Symmetry in physics has been generalized to mean
invariance Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
—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 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 PW Anderson to write in his widely read 1972 article ''More is Different'' that "it is only slightly overstating the case to say that physics is the study of symmetry." See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity 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 mathematics and geometry, 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' ...
of spacetime;
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. A family of particular transformations may be ''continuo ...
of particles; and
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
of physical theories.


In biology

In biology, the notion of symmetry is mostly used explicitly to describe body shapes.
Bilateral animals The Bilateria or bilaterians are animals with bilateral symmetry as an embryo, i.e. having a left and a right side that are mirror images of each other. This also means they have a head and a tail (anterior-posterior axis) as well as a belly and ...
, including humans, are more or less symmetric with respect to the sagittal plane 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 a group of predation, predatory marine invertebrates of the order (biology), order Actiniaria. Because of their colourful appearance, they are named after the ''Anemone'', a terrestrial flowering plant. Sea anemones are classifi ...
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 An echinoderm () is any member of the phylum Echinodermata (). The adults are recognisable by their (usually five-point) radial symmetry, and include starfish, brittle stars, sea urchins, sand dollars, and sea cucumbers, as well as the sea li ...
, the group that includes
starfish Starfish or sea stars are star-shaped echinoderms belonging to the class Asteroidea (). Common usage frequently finds these names being also applied to ophiuroids, which are correctly referred to as brittle stars or basket stars. Starfish ...
,
sea urchin Sea urchins () are 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, hard shells (tests) of ...
s, and
sea lilies Crinoids are marine animals that make up the class Crinoidea. Crinoids that are attached to the sea bottom by a stalk in their adult form are commonly called sea lilies, while the unstalked forms are called feather stars or comatulids, which are ...
. 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 to
chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
because it undergirds essentially all ''specific'' interactions between molecules in nature (i.e., via the interaction of natural and human-made chiral molecules with inherently chiral biological systems). The control of the
symmetry Symmetry (from grc, συμμετρία "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 definit ...
of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects. A rigorous understanding of symmetry explains fundamental observations in
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, and in the applied areas of
spectroscopy Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
and
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
. The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory.


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 a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mach ...
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 tradition suggested that bilateral symmetry was one of the key factors in perceptual
grouping Grouping may refer to: * Muenchian grouping * Principles of grouping * Railways Act 1921, also known as Grouping Act, a reorganisation of the British railway system * Grouping (firearms), the pattern of multiple shots from a sidearm See also ...
. 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 of
reciprocity 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, sympathy,
apology Apology, The Apology, apologize/apologise, apologist, apologetics, or apologetic may refer to: Common uses * Apology (act), an expression of remorse or regret * Apologia, a formal defense of an opinion, position, or action Arts, entertainment, ...
,
dialogue Dialogue (sometimes spelled dialog in American English) is a written or spoken conversational exchange between two or more people, and a literary and theatrical form that depicts such an exchange. As a philosophical or didactic device, it is c ...
, respect, justice, and
revenge Revenge is committing a harmful action against a person or group in response to a grievance, be it real or perceived. Francis Bacon described revenge as a kind of "wild justice" that "does... offend the law ndputteth the law out of office." Pr ...
. Reflective equilibrium is the balance that may be attained through deliberative mutual adjustment among general principles and specific judgments. Symmetrical interactions send the
moral A moral (from Latin ''morālis'') is a message that is conveyed or a lesson to be learned from a story or event. The moral may be left to the hearer, reader, or viewer to determine for themselves, or may be explicitly encapsulated in a maxim. A ...
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. Symmetrical relationships can to some degree be maintained by simple (
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
) strategies seen in
symmetric games In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff t ...
such as tit for tat.


In the arts

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


In architecture

Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothic cathedrals and The White House, through the layout of the individual floor plans, and down to the design of individual building elements such as tile mosaics.
Islam Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic religions, Abrahamic Monotheism#Islam, monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God in Islam, God (or ...
ic buildings such as the Taj Mahal and the
Lotfollah mosque Sheikh Lotfollah Mosque ( fa, مسجد شیخ لطف الله) is one of the masterpieces of Iranian architecture that was built during the Safavid Empire, standing on the eastern side of Naqsh-i Jahan Square, Esfahan, Iran. Construction of 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, Andalusia, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of the ...
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, starting with
International style International style may refer to: * International Style (architecture), the early 20th century modern movement in architecture *International style (art), the International Gothic style in medieval art *International Style (dancing), a term used in ...
, 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 ancient
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of va ...
, 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 in carpet and rug patterns spans a variety of cultures. American
Navajo The Navajo (; British English: Navaho; nv, Diné or ') are a Native American people of the Southwestern United States. With more than 399,494 enrolled tribal members , the Navajo Nation is the largest federally recognized tribe in the United ...
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 In 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 parallelogram containi ...
—that is, motifs that are reflected across both the horizontal and vertical axes (see ).


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 Beadwork is the art or craft of attaching beads to one another by stringing them onto a thread or thin wire with a sewing or beading needle or sewing them to cloth. Beads are produced in a diverse range of materials, shapes, and sizes, and vary b ...
,
furniture Furniture refers to movable objects intended to support various human activities such as seating (e.g., stools, chairs, and sofas), eating (tables), storing items, eating and/or working with an item, and sleeping (e.g., beds and hammocks). Fu ...
,
sand painting Sandpainting is the art of pouring coloured sands, and powdered pigments from minerals or crystals, or pigments from other natural or synthetic sources onto a surface to make a fixed or unfixed sand painting. Unfixed sand paintings have a long es ...
s, 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 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 music

File:Major and minor triads, triangles.png, Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (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 root of C major triad poly 502 472 522 493 502 514 560 493 root of C major triad poly 863 462 830 493 863 525 895 493 third of C major triad poly 1303 442 1160 493 1304 544 fifth of C major triad 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 music touches many aspects of the creation and perception of music.


Musical form

Symmetry has been used as a
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire ...
constraint by many composers, such as the arch (swell) form (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 in the mid to late 1960s. Reich's work is marked by its use of repetitive figures, slow harmonic rhythm, a ...
,
Béla Bartók Béla Viktor János Bartók (; ; 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 are regarded as H ...
, and James Tenney. 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 Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number ...
s and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the
diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
or the major chord. Symmetrical scales or chords, such as the whole tone scale,
augmented chord Augment or augmentation may refer to: Language *Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages *Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns i ...
, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are
ambiguous Ambiguity is the type of meaning (linguistics), meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations wikt:plausible#Adjective, plausible. A common aspect of ambiguity is uncertainty. It ...
as to the
key Key or The Key may refer to: Common meanings * Key (cryptography), a piece of information that controls the operation of a cryptography algorithm * Key (lock), device used to control access to places or facilities restricted by a lock * Key (map ...
or tonal center, and have a less specific diatonic functionality. However, composers such as
Alban Berg Alban Maria Johannes Berg ( , ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sma ...
,
Béla Bartók Béla Viktor János Bartók (; ; 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 are regarded as H ...
, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to
keys Key or The Key may refer to: Common meanings * Key (cryptography), a piece of information that controls the operation of a cryptography algorithm * Key (lock), device used to control access to places or facilities restricted by a lock * Key (map ...
or non- tonal tonal centers. George Perle explains "C–E, D–F♯, ndEb–G, are different instances of the same 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 In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of
Romantic Romantic may refer to: Genres and eras * The Romantic era, an artistic, literary, musical and intellectual movement of the 18th and 19th centuries ** Romantic music, of that era ** Romantic poetry, of that era ** Romanticism in science, of that e ...
composers such as
Gustav Mahler Gustav Mahler (; 7 July 1860 – 18 May 1911) was an Austro-Bohemian Romantic composer, and one of the leading conductors of his generation. As a composer he acted as a bridge between the 19th-century Austro-German tradition and the modernism ...
and
Richard Wagner Wilhelm Richard Wagner ( ; ; 22 May 181313 February 1883) was a German composer, theatre director, polemicist, and conductor who is chiefly known for his operas (or, as some of his mature works were later known, "music dramas"). Unlike most op ...
form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók,
Alexander Scriabin Alexander Nikolayevich Scriabin (; russian: Александр Николаевич Скрябин ; – ) was a Russian composer and virtuoso pianist. Before 1903, Scriabin was greatly influenced by the music of Frédéric Chopin and composed ...
,
Edgard Varèse Edgard Victor Achille Charles Varèse (; also spelled Edgar; December 22, 1883 – November 6, 1965) was a French-born composer who spent the greater part of his career in the United States. Varèse's music emphasizes timbre and rhythm; he coined ...
, and the Vienna school. At the same time, these progressions signal the end of tonality. The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's ''Quartet'', Op. 3 (1910).


Equivalency

Tone rows or pitch class sets which are
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under retrograde are horizontally symmetrical, under
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
vertically. See also Asymmetric rhythm.


In aesthetics

The relationship of symmetry to aesthetics is complex. Humans find
bilateral symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pla ...
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. 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 in literature, a simple example being the
palindrome A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Panam ...
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 ''Beowulf'' (; ang, Bēowulf ) is an Old English epic poem in the tradition of Germanic heroic legend consisting of 3,182 alliterative lines. It is one of the most important and most often translated works of Old English literature. The ...
''.


See also

*
Automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
*
Burnside's lemma Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when ...
*
Chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
* Even and odd functions *
Fixed points of isometry groups in Euclidean space A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space. For an object, any unique centre and, more g ...
– center of symmetry * Isotropy *
Palindrome A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Panam ...
*
Spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact s ...
* Spontaneous symmetry breaking *
Symmetry-breaking constraints In the field of mathematics called combinatorial optimization, the method of symmetry-breaking constraints can be used to take advantage of symmetries in many constraint satisfaction and optimization problems, by adding constraints that eliminate sy ...
*
Symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X( ...
* Symmetries of polyiamonds * Symmetries of polyominoes *
Symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
* Wallpaper group


Notes


References


Further reading

* ''The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry'',
Mario Livio Mario Livio (born June 19, 1945) is an Israeli-American astrophysicist and an author of works that popularize science and mathematics. For 24 years (1991-2015) he was an astrophysicist at the Space Telescope Science Institute, which operates th ...
,
Souvenir Press Ernest Hecht (21 September 1929 – 13 February 2018)Katherine Cowdrey"'Wise and witty' Ernest Hecht dies, aged 88" ''The Bookseller'', 13 February 2018. was a British publisher, producer, and philanthropist. In 1951, he founded Souvenir Press L ...
2006,


External links


International Symmetry Association (ISA)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 Geometry Theoretical physics Artistic techniques Aesthetics