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 definition, and is usually used to refer to an object that is Invariant (mathematics), invariant under some Transformation (function), transformations; including Translation (geometry), translation, Reflection (mathematics), reflection, Rotation (mathematics), rotation or Scaling (geometry), scaling. 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 space, spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including scientific model, theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music. The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.

In mathematics

In geometry

The armoured triskelion on the flag of the Isle of Man

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 (line or mirror symmetry) if there is a line (or in three-dimensional space, 3D a plane) going through it which divides it into two pieces that are mirror images of each other. *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 translation (geometry), translated (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. *An object has scale symmetry 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 similarity (geometry), similar in shape to larger portions. *Other symmetries include glide reflection symmetry (a reflection followed by a translation) and improper rotation, rotoreflection symmetry (a combination of a rotation and a reflection).

In logic

A binary relation, dyadic relation ''R'' = ''S'' × ''S'' is symmetric if for all elements ''a'', ''b'' in ''S'', whenever it is true that ''Rab'', it is also true that ''Rba''. 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 ''logical conjunction, and'' (∧, or &), ''logical disjunction, or'' (∨, or , ) and ''if and only if'' (↔), while the connective ''if'' (→) is not symmetric. Other symmetric logical connectives include ''logical nand, nand'' (not-and, or ⊼), ''xor'' (not-biconditional, or ⊻), and ''logical nor, nor'' (not-or, or ⊽).

Other areas of mathematics

Generalizing from geometrical symmetry in the previous section, one can say that a mathematical object is ''symmetric'' with respect to a given Operation (mathematics), mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group (mathematics), group. In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrix, symmetric matrices in linear algebra, and Galois groups in Galois theory. In statistics, symmetry also manifests as symmetric probability distributions, and as skewness—the asymmetry of distributions.

In science and nature

In physics

Symmetry in physics has been generalized to mean Invariant (physics), invariance—that is, lack of change—under any kind of transformation, for example General covariance, arbitrary coordinate transformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate Philip Warren Anderson, PW Anderson to write in his widely read 1972 article ''More is Different'' that "it is only slightly overstating the case to say that physics is the study of symmetry." 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 symmetry, continuous symmetries and discrete symmetry, discrete symmetries of spacetime; internal symmetry, internal symmetries of particles; and supersymmetry of physical theories.

In biology

Studio del Corpo Umano - Leonardo da Vinci

In biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateria, Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The cephalisation, head becomes specialized with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric. Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea 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

Symmetry is important to chemistry because it undergirds essentially all ''specific'' interactions between molecules in nature (i.e., via the interaction of natural and human-made chiral (chemistry), chiral molecules with inherently chiral biological systems). The control of the molecular symmetry, symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer drug, therapeutic interventions with minimal side effects. A rigorous understanding of symmetry explains fundamental observations in quantum chemistry, and in the applied areas of spectroscopy and crystallography. The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory.

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 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 psychology, Gestalt tradition suggested that bilateral symmetry was one of the key factors in perceptual Principles of grouping, grouping. This is known as the Gestalt psychology#Law of Symmetry, Law of Symmetry. The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry is faster when this is a property of a single object. 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 (social psychology), reciprocity, empathy, sympathy, Remorse, apology, dialogue, respect, justice, and revenge. Reflective equilibrium is the balance that may be attained through deliberative mutual adjustment among general principles and specific judgments. Symmetrical interactions send the morality, moral message "we are all the same" while asymmetrical interactions may send the message "I am special; better than you." Peer relationships, such as can be governed by the golden rule, are based on symmetry, whereas power relationships are based on asymmetry. Symmetrical relationships can to some degree be maintained by simple (game theory) strategies seen in symmetric games such as tit for tat.

In the arts

Isfahan Lotfollah mosque ceiling symmetric

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

In architecture

Great Mosque of Kairouan, west portico of the courtyard

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 mosaic, tile mosaics. Islamic buildings such as the Taj Mahal and the Lotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation. Moorish buildings like the Alhambra 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 (architecture), 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 ancient Chinese people, Chinese, for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.

In carpets and rugs

A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo people, Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of a rectangle—that is, Motif (visual arts), motifs that are reflected across both the horizontal and vertical axes (see ).

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 music

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

Musical form

Symmetry has been used as a musical form, formal constraint by many composers, such as the arch form, arch (swell) form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney. In classical music, Bach used the symmetry concepts of permutation and invariance.

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, 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 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. 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 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''.

Notes

References

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 Symmetry, Geometry Theoretical physics Artistic techniques Aesthetics