Symmetry (Ancient Greek συμμετρία 'agreement in dimensions, due proportion, arrangement') in everyday life refers to a sense of harmonious and beautiful proportion and balance. 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.
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
objects, including theoretic models, language, and music.
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.
In logic
A dyadic relation R = S × S is symmetric if for all elements a, b in S,
whenever it is true that Rab, it is also true that Rba. Thus, the relation
"is the same age as" is symmetric, for if Paul is the same age as
Mary, then Mary is the same age as Paul.
In physics
Symmetry in physics has been generalized to mean invariance—that is,
lack of change—under any kind of transformation, for example arbitrary
coordinate transformations. This concept has become one of the most powerful
tools of theoretical physics, as it has become evident that practically all
laws of nature originate in symmetries.
Important symmetries in physics include continuous symmetries and
discrete symmetries of spacetime; internal symmetries of particles; and
supersymmetry of physical theories.
In biology
In biology, the notion of symmetry is mostly used explicitly to describe
body shapes. Bilateral animals, including humans, are more or less symmetric
with respect to the sagittal plane which divides the body into left and right
halves. 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 molecules with inherently chiral biological
systems). The control of the symmetry of molecules produced in modern chemical
synthesis contributes to the ability of scientists to offer therapeutic
interventions with minimal side effects.
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. Both behavioural and neurophysiological studies have
confirmed the special sensitivity to reflection symmetry in humans and also in
other animals. 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, empathy, sympathy, 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 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.
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. Islamic buildings such as
the Taj Mahal and the Lotfollah mosque make elaborate use of symmetry both in
their structure and in their ornamentation.
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.
In carpets and rugs
A long tradition of the use of symmetry in carpet and rug patterns spans
a variety of cultures. American 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, motifs that are reflected
across both the horizontal and vertical axes.
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 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.
In music
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.
Symmetry is also an important consideration in the formation of scales
and chords, traditional or tonal music being made up of non-symmetrical groups
of pitches, such as the diatonic scale or the major chord.
In aesthetics
The relationship of symmetry to aesthetics is complex. Humans find
bilateral symmetry in faces physically attractive; it indicates health and
genetic fitness. 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.
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