Symmetry (from Greek συμμετρία symmetria "agreement in
dimensions, due proportion, arrangement") in everyday language
refers to a sense of harmonious and beautiful proportion and
balance.[a] In mathematics, "symmetry" has a more precise
definition, that an object is invariant to any of various
transformations; including reflection, rotation or scaling. Although
these two meanings of "symmetry" can sometimes be told apart, they are
related, so in this article they 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, theoretic models, language, music and even knowledge
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.
1 In mathematics
1.1 In geometry
1.2 In logic
1.3 Other areas of mathematics
2 In science and nature
2.1 In physics
2.2 In biology
2.3 In chemistry
3 In social interactions
4 In the arts
4.1 In architecture
4.2 In pottery and metal vessels
4.3 In quilts
4.4 In carpets and rugs
4.5 In music
4.5.1 Musical form
4.5.2 Pitch structures
4.6 In other arts and crafts
4.7 In aesthetics
4.8 In literature
5 See also
8 Further reading
9 External links
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.
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 going through it which divides it into two pieces which are
mirror images of each other.
An object has rotational symmetry if the object can be rotated about a
fixed point without changing the overall shape.
An object has translational symmetry if it can be translated 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
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 small portions of the fractal are similar in shape to
Other symmetries include glide reflection symmetry and rotoreflection
A dyadic relation R is symmetric if and only if, whenever it's true
that Rab, it's true that Rba. Thus, "is the same age as" is
symmetrical, for if Paul is the same age as Mary, then Mary is the
same age as Paul.
Symmetric binary logical connectives are and (∧, or &), or (∨,
or ), biconditional (if and only if) (↔), nand (not-and, or ⊼),
xor (not-biconditional, or ⊻), and nor (not-or, or ⊽).
Other areas of mathematics
Generalizing from geometrical symmetry in the previous section, we 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;
the symmetric group in abstract algebra; symmetric matrices in linear
algebra; and the
Galois group in Galois theory. In statistics, it
appears as symmetric probability distributions, and as skewness,
asymmetry of distributions.
In science and nature
Further information: Patterns in nature
Symmetry in physics
Symmetry 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.
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
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 of spacetime; internal symmetries of particles;
and supersymmetry of physical theories.
Many animals are approximately mirror-symmetric, though internal
organs are often arranged asymmetrically.
Further information: symmetry in biology and facial symmetry
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. 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 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
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
Main article: molecular symmetry
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. 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 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,
dialog, 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. 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
The ceiling of Lotfollah mosque, Isfahan,
Iran has 8-fold symmetries.
Mathematics and art
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 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. Moorish buildings like the
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, relies instead on "wings and
balance of masses".
In pottery and metal vessels
Clay pots thrown on a 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
Kitchen Kaleidoscope 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
In carpets and rugs
Persian rug with rectangular symmetry
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 (see Klein four-group
Major and minor triads on the white piano keys are symmetrical to the
D. (compare article) (file)
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.
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. 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 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 keys or
non-tonal tonal centers. explains "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:"
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 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.
The first extended composition consistently based on symmetrical pitch
relations was probably Alban Berg's Quartet, Op. 3 (1910).
Tone rows or pitch class sets which are invariant under retrograde are
horizontally symmetrical, under inversion vertically. See also
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, batik, ikat, carpet-making, and many
kinds of textile and embroidery patterns.
Symmetry (physical attractiveness)
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.
People prefer shapes that have some symmetry, but enough complexity to
make them interesting.
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, as in the
rise:fall pattern of Beowulf.
Even and odd functions
Fixed points of isometry groups in Euclidean space – center of
Spontaneous symmetry breaking
Symmetries of polyiamonds
Symmetries of polyominoes
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Look up symmetry in Wiktionary, the free dictionary.
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