Symmetry

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This article is about the broad concept. For other uses, see Symmetry (disambiguation).
Symmetry (left) and asymmetry (right)
A spherical symmetry group with octahedral symmetry. The yellow region shows the fundamental domain.
reflectional symmetry, rotational symmetry and self-similarity, three forms of symmetry. This shape is obtained by a finite subdivision rule
.
plane
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Symmetry (from

Ancient Greek συμμετρία (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] In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling
. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.

Mathematical symmetry may be observed with respect to the passage of

theoretic models, language, and music.[4][b]

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 of symmetry.

In mathematics

In geometry

Main article: Symmetry (geometry)
The triskelion has 3-fold rotational symmetry.

A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.[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 has
    reflectional symmetry (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.[6]
  • 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.[7]
  • An object has translational symmetry if it can be translated (moving every point of the object by the same distance) without changing its overall shape.[8]
  • 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.[9]
  • An object has
    Fractals also exhibit a form of scale symmetry, where smaller portions of the fractal are similar in shape to larger portions.[11]
  • Other symmetries include glide reflection symmetry (a reflection followed by a translation) and rotoreflection symmetry (a combination of a rotation and a reflection[12]).

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.[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 binary

nor
(not-or, or ⊽).

Other areas of mathematics

Main article: Symmetry in 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.[15] 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, symmetry also manifests as symmetric probability distributions, and as skewness—the asymmetry of distributions.[16]

In science and nature

Further information: Patterns in nature

In physics

Main article:
Symmetry in physics

Symmetry in physics has been generalized to mean

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."[18] 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);[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]

Important symmetries in physics include

internal symmetries of particles; and supersymmetry
of physical theories.

In biology

Further information: symmetry in biology and facial symmetry
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.

In biology, the notion of symmetry is mostly used explicitly to describe body shapes.

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.[22]

Plants and sessile (attached) animals such as

sea lilies.[23]

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]

In chemistry

Main article: Molecular symmetry

Symmetry is important to

physical science draws heavily on the mathematical area of group theory.[26]

In psychology and neuroscience

Further information: Visual perception

For a human observer, some symmetry types are more salient than others, in particular the most salient is a reflection with a vertical axis, like that present in the human face. Ernst 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 the Gestalt 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.[29] 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.[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]

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, empathy, sympathy, apology, dialogue, respect, justice, and revenge.

judgments.[34]
Symmetrical interactions send the
symmetric games such as tit for tat.[36]

In the arts

Further information: Mathematics and art

There exists a list of journals and newsletters known to deal, at least in part, with symmetry and the arts.[37]

In architecture

Further information: Mathematics and architecture
Seen from the side, the Taj 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 Gothic

Lotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation.[38][39] Moorish buildings like the Alhambra 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";

International style, relies instead on "wings and balance of masses".[41]

In pottery and metal vessels

pottery wheel
acquire rotational symmetry.

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, 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]

In carpets and rugs

Persian rug with rectangular symmetry

A long tradition of the use of symmetry in

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, motifs that are reflected across both the horizontal and vertical axes (see Klein four-group § Geometry).[43][44]

In quilts

Kitchen kaleidoscope quilt block

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.[45]

In other arts and crafts

Further information: Islamic geometric patterns

Symmetries appear in the design of objects of all kinds. Examples include

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.[46]

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.

In music

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

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 constraint by many composers, such as the 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.[48]

Pitch structures

Symmetry is also an important consideration in the formation of

keys or non-tonal tonal centers.[49] George Perle explains that "C–E, D–F♯, [and] Eb–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:"[49]

D D♯ E F F♯ G G♯
D C♯ C B A♯ A G♯

Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).[49]

+ 2 3 4 5 6 7 8
2 1 0 11 10 9 8
4 4 4 4 4 4 4

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are

enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of 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.[49][50]

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910).[50]

Equivalency

Asymmetric rhythm
.

In aesthetics

Main article:
Symmetry (physical attractiveness)

The relationship of symmetry to

bilateral 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]

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.[55]

See also

Explanatory notes

  1. ^ 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.
  2. mathematical equation
    or a series of tones (music).

References

  1. ^ Harper, Douglas. "symmetry". Online Etymology Dictionary.
  2. ISBN 978-0-691-13482-6
    .
  3. ^ Hill, C. T.; Lederman, L. M. (2005). Symmetry and the Beautiful Universe. Prometheus Books.
  4. ISBN 981-256-192-7
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  5. ^ E. H. Lockwood, R. H. Macmillan, Geometric Symmetry, London: Cambridge Press, 1978
  6. ISBN 0-691-02374-3
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  7. ^ Singer, David A. (1998). Geometry: Plane and Fancy. Springer Science & Business Media.
  8. ^ Stenger, Victor J. (2000) and Mahou Shiro (2007). Timeless Reality. Prometheus Books. Especially chapter 12. Nontechnical.
  9. ^ Bottema, O, and B. Roth, Theoretical Kinematics, Dover Publications (September 1990)
  10. ^ Tian Yu Cao Conceptual Foundations of Quantum Field Theory Cambridge University Press p.154-155
  11. ISBN 978-0-387-94153-0
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  12. ^ "Rotoreflection Axis". TheFreeDictionary.com. Retrieved 2019-11-12.
  13. ^ Josiah Royce, Ignas K. Skrupskelis (2005) The Basic Writings of Josiah Royce: Logic, loyalty, and community (Google eBook) Fordham Univ Press, p. 790
  14. ^ Gao, Alice (2019). "Propositional Logic: Introduction and Syntax" (PDF). University of Waterloo — School of Computer Science. Retrieved 2019-11-12.
  15. ^ Christopher G. Morris (1992) Academic Press Dictionary of Science and Technology Gulf Professional Publishing
  16. doi:10.3390/e5030271
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  17. ^ 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.
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  20. S2CID 121773411
  21. ^ Valentine, James W. "Bilateria". AccessScience. Archived from the original on 18 January 2008. Retrieved 29 May 2013.
  22. ^ Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002). "Animal Diversity (Third Edition)" (PDF). Chapter 8: Acoelomate Bilateral Animals. McGraw-Hill. p. 139. Archived from the original (PDF) on May 17, 2016. Retrieved October 25, 2012.
  23. ^ Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. pp. 64–65.
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  26. ISBN 0-12-457551-X
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  27. ^ Mach, Ernst (1897). Symmetries and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Open Court Publishing House.
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  34. ^ Daniels, Norman (2003-04-28). "Reflective Equilibrium". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  35. ^ Emotional Competency: Symmetry
  36. ^ Lutus, P. (2008). "The Symmetry Principle". Retrieved 28 September 2015.
  37. doi:10.18713/JIMIS-230718-4-1
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  38. ^ Williams: Symmetry in Architecture. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.
  39. ^ Aslaksen: Mathematics in Art and Architecture. Math.nus.edu.sg. Retrieved on 2013-04-16.
  40. ISBN 978-1-4008-2311-6
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  41. ^ a b Dunlap, David W. (31 July 2009). "Behind the Scenes: Edgar Martins Speaks". New York Times. Retrieved 11 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.'
  42. ^ The Art of Chinese Bronzes Archived 2003-12-11 at the Wayback Machine. Chinavoc (2007-11-19). Retrieved on 2013-04-16.
  43. ^ Marla Mallett Textiles & Tribal Oriental Rugs. The Metropolitan Museum of Art, New York.
  44. ^ Dilucchio: Navajo Rugs. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.
  45. ^ Quate: Exploring Geometry Through Quilts Archived 2003-12-31 at the Wayback Machine. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.
  46. ISBN 978-0-521-72876-8
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  47. ^ "How to Design a Perfect Logo with Grid and Symmetry".
  48. ^ see ("Fugue No. 21," pdf Archived 2005-09-13 at the Wayback Machine or Shockwave Archived 2005-10-26 at the Wayback Machine)
  49. ^
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  50. ^
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  51. S2CID 1205083
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  52. ISBN 1-56750-636-4
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  53. ^ 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.
  54. ^ Arnheim, Rudolf (1969). Visual Thinking. University of California Press.
  55. ^ Jenny Lea Bowman (2009). "Symmetrical Aesthetics of Beowulf". University of Tennessee, Knoxville.

Further reading

  • The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry,
    ISBN 0-285-63743-6

External links

Look up symmetry in Wiktionary, the free dictionary.
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