"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, and is usually used to refer to an object that is invariant
under some transformations
; including translation
Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
The opposite of symmetry is asymmetry
, which refers to the absence or a violation 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.
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.
- 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 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 similar in shape to larger portions.
- Other symmetries include glide reflection symmetry (a reflection followed by a translation) and rotoreflection symmetry (a combination of a rotation and a reflection).
A dyadic relation R
is symmetric if for each element a
, 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
(∧, or &), or
(∨, or |) and if and only if
(↔), while the connective if
(→) is not symmetric.
Other symmetric logical connectives include 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, one can say that a mathematical object
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 science and nature
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 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.
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Leonardo da Vinci
's 'Vitruvian Man
' (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.
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.
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.
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
. 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
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.
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
, 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
It exists a list of journals and newsletters known to deal, at least for a part, with symmetry and arts.
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
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.
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
, 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
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 § Geometry
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 is also an important consideration in the formation of scales
, 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
George Perle 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
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
, sand paintings
, 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 decoration
, carpet-making, and many kinds of textile
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.
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
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