Symmetry (from Ancient Greekσυμμετρία (summetría)'agreement in dimensions, due proportion, arrangement')[1] in everyday life refers to a sense of harmonious and beautiful proportion and balance.[2][3][a] Inmathematics, the term has a more precise definition and is usually used to refer to an object that isinvariant under sometransformations, such astranslation,reflection,rotation, orscaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
This article describes symmetry from three perspectives: inmathematics, includinggeometry, the most familiar type of symmetry for many people; inscience andnature; and in the arts, coveringarchitecture,art, and music.
The opposite of symmetry isasymmetry, which refers to the absence of symmetry.
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.[5] This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
An object hasreflectional symmetry (line or mirror symmetry) if there is a line (or in3D a plane) going through it which divides it into two pieces that are mirror images of each other.[6]
An object hasrotational symmetry if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.[7]
An object hastranslational symmetry if it can betranslated (moving every point of the object by the same distance) without changing its overall shape.[8]
An object hashelical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as ascrew axis.[9]
An object hasscale symmetry if it does not change shape when it is expanded or contracted.[10]Fractals also exhibit a form of scale symmetry, where smaller portions of the fractal aresimilar in shape to larger portions.[11]
Other symmetries includeglide reflection symmetry (a reflection followed by a translation) androtoreflection symmetry (a combination of a rotation and a reflection[12]).
Adyadic relationR =S ×S is symmetric if for all elementsa,b inS, whenever it is true thatRab, it is also true thatRba.[13] 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 connectives includeand (∧, or &),or (∨, or |) andif and only if (↔), while the connectiveif (→) is not symmetric.[14] Other symmetric logical connectives includenand (not-and, or ⊼),xor (not-biconditional, or ⊻), andnor (not-or, or ⊽).
Generalizing from geometrical symmetry in the previous section, one can say that amathematical object issymmetric with respect to a givenmathematical operation, if, when applied to the object, this operation preserves some property of the object.[15] The set of operations that preserve a given property of the object form agroup.
Symmetry in physics has been generalized to meaninvariance—that is, lack of change—under any kind of transformation, for examplearbitrary coordinate transformations.[17] This concept has become one of the most powerful tools oftheoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureatePW Anderson to write in his widely read 1972 articleMore is Different that "it is only slightly overstating the case to say that physics is the study of symmetry."[18] SeeNoether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language);[19] and also,Wigner's classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.[20]
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
In biology, the notion of symmetry is mostly used explicitly to describe body shapes.Bilateral animals, including humans, are more or less symmetric with respect to thesagittal plane which divides the body into left and right halves.[21] Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. Thehead becomes 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.[22]
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.[24][25]
Symmetry is important tochemistry because it undergirds essentially allspecific interactions between molecules in nature (i.e., via the interaction of natural and human-madechiral molecules with inherently chiral biological systems). The control of thesymmetry of molecules produced in modernchemical synthesis contributes to the ability of scientists to offertherapeutic interventions with minimalside effects. A rigorous understanding of symmetry explains fundamental observations inquantum chemistry, and in the applied areas ofspectroscopy andcrystallography. The theory and application of symmetry to these areas ofphysical science draws heavily on the mathematical area ofgroup theory.[26]
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),[27] 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.[28] Early studies within theGestalt tradition suggested that bilateral symmetry was one of the key factors in perceptualgrouping. This is known as theLaw of Symmetry. The role of symmetry in grouping and figure/ground organization has been confirmed in many studies. For instance, detection of reflectional symmetry is faster when this is a property of a single object.[29] Studies of human perception andpsychophysics 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.[30]
More recent neuroimaging studies have documented which brain regions are active during perception of symmetry. Sasaki et al.[31] 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.[32] 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.[33]
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments ofreciprocity,empathy,sympathy,apology,dialogue, respect,justice, andrevenge.Reflective equilibrium is the balance that may be attained through deliberative mutual adjustment among general principles and specificjudgments.[34]Symmetrical interactions send themoral 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 theGolden Rule, are based on symmetry, whereas power relationships are based on asymmetry.[35] Symmetrical relationships can to some degree be maintained by simple (game theory) strategies seen insymmetric games such astit for tat.[36]
Seen from the side, theTaj Mahal has bilateral symmetry; from the top (in plan), it has fourfold symmetry.
Symmetry finds its ways into architecture at every scale, from the overall external views of buildings such as Gothiccathedrals andThe White House, through the layout of the individualfloor plans, and down to the design of individual building elements such astile mosaics.Islamic buildings such as theTaj Mahal and theLotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation.[38][39] Moorish buildings like theAlhambra are ornamented with complex patterns made using translational and reflection symmetries as well as rotations.[40]
It has been said that only bad architects rely on a "symmetrical layout of blocks, masses and structures";[41]Modernist architecture, starting withInternational style, relies instead on "wings and balance of masses".[41]
Clay pots thrown on apottery wheel acquire rotational symmetry.
Since the earliest uses ofpottery 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, 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.[42]
A long tradition of the use of symmetry incarpet and rug patterns spans a variety of cultures. AmericanNavajo Indians used bold diagonals and rectangular motifs. ManyOriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs have typically the symmetries of arectangle—that is,motifs that are reflected across both the horizontal and vertical axes (seeKlein four-group § Geometry).[43][44]
Asquilts 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.[45]
Symmetry is also used in designing logos.[47] 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.
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).[49]
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Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which areenharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonalprogressions in the works ofRomantic composers such asGustav Mahler andRichard 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.[49][50]
The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg'sQuartet, Op. 3 (1910).[50]
The relationship of symmetry toaesthetics is complex. Humans findbilateral symmetry in faces physically attractive;[51] it indicates health and genetic fitness.[52][53] 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.[54]
Symmetry can be found in various forms inliterature, a simple example being thepalindrome where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, such as the rise and fall pattern ofBeowulf.[55]
^For example,Aristotle ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.
^Costa, Giovanni; Fogli, Gianluigi (2012).Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Springer Science & Business Media. p. 112.
^Mach, Ernst (1897).Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Open Court Publishing House.
^abDunlap, David W. (31 July 2009)."Behind the Scenes: Edgar Martins Speaks".New York Times. Retrieved11 November 2014."My starting point for this construction was a simple statement which I once read (and which does not necessarily reflect my personal views): 'Only a bad architect relies on symmetry; instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses.'
^Cucker, Felipe (2013).Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University Press. pp. 77–78, 83, 89, 103.ISBN978-0-521-72876-8.
^Grammer, K.; Thornhill, R. (1994). "Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness".Journal of Comparative Psychology.108 (3). Washington, D.C.:233–42.doi:10.1037/0735-7036.108.3.233.PMID7924253.S2CID1205083.
^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.
^Arnheim, Rudolf (1969).Visual Thinking. University of California Press.
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