# Dictionary Definition

symmetry

### Noun

1 (mathematics) an attribute of a shape or
relation; exact correspondence of form on opposite sides of a
dividing line or plane [syn: symmetricalness,
correspondence,
balance] [ant: asymmetry]

2 balance among the parts of something [syn:
proportion] [ant:
disproportion]

3 (physics) the property of being isotropic;
having the same value when measured in different directions [syn:
isotropy] [ant:
anisotropy]

# User Contributed Dictionary

## English

### Etymology

symmetria, originally from sc=polytonic < | + |; i.e. like measure or similar measurement.### Noun

- Exact correspondence on either side of a dividing line, plane, center or axis.
- The satisfying arrangement of a balanced distribution of the elements of a whole.

#### Related terms

#### Translations

correspondence on either side of a dividing
line, plane, center or axis

satisfying arrangement of a balanced
distribution of the elements of a whole.

- Bosnian: simetrija
- Croatian: simetrija
- Serbian: simetrija
- Slovene: simetrija

# Extensive Definition

Symmetry generally conveys two primary meanings.
The first is an imprecise sense of harmonious or
aesthetically-pleasing proportionality and balance; such that it
reflects beauty or perfection. The second meaning is a precise and
well-defined concept of balance or "patterned self-similarity" that
can be demonstrated or proved according to the rules of a formal
system: by geometry, through physics or otherwise.

Although the meanings are distinguishable, in
some contexts, both meanings of "symmetry" are related and
discussed in parallel.

The "precise" notions of symmetry have various
measures and operational definitions. For example, symmetry may be
observed:

- with respect to the passage of time;
- as a spatial relationship;
- through geometric transformations such as scaling, reflection, and rotation;
- through other kinds of functional transformations; and
- as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.

This article describes these notions of symmetry
from three perspectives. The first is that of mathematics, in which
symmetries are defined and categorized precisely. The second
perspective describes symmetry as it relates to science and technology. In this context,
symmetries underlie some of the most profound results of modern
physics, including
aspects of space and
time. Finally, a third perspective discusses symmetry in the
humanities, covering
its rich and varied use in history, architecture, art, and religion.

The opposite of symmetry is asymmetry.

## Symmetry in the field of mathematics

In formal terms, we say that an object is
symmetric with respect to a given mathematical
operation, if, when applied to the object, this operation does
not change the object or its appearance. Two objects are symmetric
to each other with respect to a given group of operations if one is
obtained from the other by some of the operations (and vice
versa).

Symmetries may also be found in living organisms
including humans and other animals (see symmetry
in biology below). In 2D geometry the main kinds of symmetry of
interest are with respect to the basic Euclidean
plane isometries: translations,
rotations, reflections,
and glide
reflections.

### Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetrical to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: V → V acting on the set of functions x: V → W by (gx)(v)=x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v)=x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v)=w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself.In a modified version for vector
fields, we have (gx)(v)=h(g,x(g−1(v))) where h
rotates any vectors and pseudovectors in x, and inverts any vectors
(but not pseudovectors) according to rotation and inversion in g,
see symmetry
in physics. The symmetry group of x consists of all g for which
x(v)=h(g,x(g(v))) for all v. In this case the symmetry group of a
constant function may be a proper subgroup of G: a constant vector
has only rotational symmetry with respect to rotation about an axis
if that axis is in the direction of the vector, and only inversion
symmetry if it is zero.

For a common notion of symmetry in Euclidean
space, G is the Euclidean
group E(n), the group of isometries, and V is the
Euclidean space. The rotation group of an object is the symmetry
group if G is restricted to E+(n), the group of direct isometries.
(For generalizations, see the next subsection.) Objects can be
modeled as functions x, of which a value may represent a selection
of properties such as color, density, chemical composition, etc.
Depending on the selection we consider just symmetries of sets of
points (x is just a boolean function of position v),
or, at the other extreme, e.g. symmetry of right and left hand with
all their structure.

For a given symmetry group, the properties of
part of the object, fully define the whole object. Considering
points equivalent
which, due to the symmetry, have the same properties, the equivalence
classes are the
orbits of the group action on the space itself. We need the
value of x at one point in every orbit to define the full object. A
set of such representatives forms a fundamental
domain. The smallest fundamental domain does not have a
symmetry; in this sense, one can say that symmetry relies upon
asymmetry.

An object with a desired symmetry can be produced
by choosing for every orbit a single function value. Starting from
a given object x we can e.g.:

- take the values in a fundamental domain (i.e., add copies of the object)

- take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap)

If it is desired to have no more symmetry than
that in the symmetry group, then the object to be copied should be
asymmetric.

As pointed out above, some groups of isometries
are not the symmetry group of any object, except in the modified
model for vector fields. For example, this applies in 1D for the
group of all translations. The fundamental domain is only one
point, so we can not make it asymmetric, so any "pattern" invariant
under translation is also invariant under reflection (these are the
uniform "patterns").

In the vector field version continuous
translational symmetry does not imply reflectional symmetry: the
function value is constant, but if it contains nonzero vectors,
there is no reflectional symmetry. If there is also reflectional
symmetry, the constant function value contains no nonzero vectors,
but it may contain nonzero pseudovectors. A corresponding 3D
example is an infinite cylinder
with a current perpendicular to the axis; the magnetic
field (a pseudovector) is, in the
direction of the cylinder, constant, but nonzero. For vectors (in
particular the current
density) we have symmetry in every plane perpendicular to the
cylinder, as well as cylindrical symmetry. This cylindrical
symmetry without mirror planes through the axis is also only
possible in the vector field version of the symmetry concept. A
similar example is a cylinder rotating about its axis, where
magnetic field and current density are replaced by angular
momentum and velocity, respectively.

A symmetry group is said to act transitively on a
repeated feature of an object if, for every pair of occurrences of
the feature there is a symmetry operation mapping the first to the
second. For example, in 1D, the symmetry group of acts transitively
on all these points, while does not act transitively on all points.
Equivalently, the first set is only one conjugacy
class with respect to isometries, while the second has two
classes.

### Non-isometric symmetry

As mentioned above, G (the symmetry group of the
space itself) may differ from the Euclidean
group, the group of isometries.

Examples:

- G is the group of similarity transformations, i.e. affine transformations with a matrix A that is a scalar times an orthogonal matrix. Thus dilations are added, self-similarity is considered a symmetry

- G is the group of affine transformations with a matrix A with determinant 1 or -1, i.e. the transformation which preserve area; this adds e.g. oblique reflection symmetry.

- G is the group of all bijective affine transformations

- In inversive geometry, G includes circle reflections, etc.

- More generally, an involution defines a symmetry with respect to that involution.

### Directional symmetry

### Reflection symmetry

Reflection symmetry, mirror symmetry,
mirror-image symmetry, or bilateral symmetry is symmetry with
respect to reflection.

In 1D, there is a point of symmetry. In 2D there
is an axis of symmetry, in 3D a plane of symmetry. An object or
figure which is indistinguishable from its transformed image is
called mirror symmetric (see mirror
image).

The axis of symmetry of a two-dimensional figure
is a line such that, if a perpendicular is constructed, any two
points lying on the perpendicular at equal distances from the axis
of symmetry are identical. Another way to think about it is that if
the shape were to be folded in half over the axis, the two halves
would be identical: the two halves are each other's mirror image.
Thus a square
has four axes of symmetry, because there are four different ways to
fold it and have the edges all match. A circle has infinitely many axes
of symmetry, for the same reason.

If the letter T is reflected along a vertical
axis, it appears the same. Note that this is sometimes called
horizontal symmetry, and sometimes vertical symmetry! One can
better use an unambiguous formulation, e.g. "T has a vertical
symmetry axis."

The triangles with this symmetry
are isosceles, the
quadrilaterals
with this symmetry are the kites and
the isosceles trapezoids.

For each line or plane of reflection, the
symmetry group is isomorphic with Cs (see point groups
in three dimensions), one of the three types of order two
(involutions), hence algebraically C2. The fundamental domain is a
half-plane or half-space.

Bilateria
(bilateral animals, including humans) are more or less symmetric
with respect to the sagittal
plane.

In certain contexts there is rotational symmetry
anyway. Then mirror-image symmetry is equivalent with inversion
symmetry; in such contexts in modern physics the term P-symmetry is
used for both (P stands for parity).

For more general types of reflection there are
corresponding more general types of reflection symmetry.
Examples:

- with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc).

- with respect to circle inversion

### Rotational symmetry

Rotational symmetry is symmetry with respect to
some or all rotations in m-dimensional Euclidean space. Rotations
are direct isometries, i.e., isometries preserving orientation.
Therefore a symmetry group of rotational symmetry is a subgroup of
E+(m) (see Euclidean
group).

Symmetry with respect to all rotations about all
points implies translational symmetry with respect to all
translations, and the symmetry group is the whole E+(m). This does
not apply for objects because it makes space homogeneous, but it
may apply for physical laws.

For symmetry with respect to rotations about a
point we can take that point as origin. These rotations form the
special
orthogonal group SO(m), the group of m×m orthogonal
matrices with determinant 1. For m=3 this
is the rotation
group.

In another meaning of the word, the rotation
group of an object is the symmetry group within E+(n), the group of
direct isometries; in other words, the intersection of the full
symmetry group and the group of direct isometries. For chiral
objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do
not distinguish different directions in space. Because of Noether's
theorem, rotational symmetry of a physical system is equivalent
to the angular momentum conservation law. See also rotational
invariance.

### Translational symmetry

See main article translational
symmetry.

Translational symmetry leaves an object invariant
under a discrete or continuous group of translations
T_a(p)=p+a

### Glide reflection symmetry

A glide
reflection symmetry (in 3D: a glide plane symmetry) means that
a reflection in a line or plane combined with a translation along
the line / in the plane, results in the same object. It implies
translational symmetry with twice the translation vector.

The symmetry group is isomorphic with Z.

### Rotoreflection symmetry

In 3D, rotoreflection or improper
rotation in the strict sense is rotation about an axis,
combined with reflection in a plane perpendicular to that axis. As
symmetry groups with regard to a roto-reflection we can
distinguish:

- the angle has no common divisor with 360°, the symmetry group is not discrete
- 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n=1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
- Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.

### Helical symmetry

Helical symmetry is
the kind of symmetry seen in such everyday objects as springs,
Slinky toys,
drill
bits, and augers. It
can be thought of as rotational symmetry along with translation
along the axis of rotation, the screw axis).
The concept of helical symmetry can be visualized as the tracing in
three-dimensional space that results from rotating an object at an
even angular
speed while simultaneously moving at another even speed along
its axis of rotation (translation). At any one point in time, these
two motions combine to give a coiling angle that helps define the
properties of the tracing. When the tracing object rotates quickly
and translates slowly, the coiling angle will be close to 0°.
Conversely, if the rotation is slow and the translation speedy, the
coiling angle will approach 90°.

Three main classes of helical symmetry can be
distinguished based on the interplay of the angle of coiling and
translation symmetries along the axis:

- Infinite helical symmetry. If there are no distinguishing features along the length of a helix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a cross section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.

- n-fold helical symmetry. If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle \theta and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called n-fold helical symmetry, where n = 360°/\theta, see e.g. double helix. This concept can be further generalized to include cases where m\theta is a multiple of 360°—that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.

- Non-repeating helical symmetry. This is the case in which the angle of rotation \theta required to observe the symmetry is an irrational number such as \sqrt 2 radians that never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating point group in two dimensions. DNA is an example of this type of non-repeating helical symmetry.

### Scale symmetry and fractals

Scale symmetry refers to the idea that if an
object is expanded or reduced in size, the new object has the same
properties as the original. Scale symmetry is notable for the fact
that it does not exist for most physical systems, a point that was
first discerned by Galileo. Simple
examples of the lack of scale symmetry in the physical world
include the difference in the strength and size of the legs of
elephants versus
mice, and the observation
that if a candle made of soft wax was enlarged to the size of a
tall tree, it would immediately collapse under its own
weight.

A more subtle form of scale symmetry is
demonstrated by fractals. As conceived by
Mandelbrot,
fractals are a mathematical concept in which the structure of a
complex form looks exactly the same no matter what degree of
magnification is
used to examine it. A coast is an example of a naturally
occurring fractal, since it retains roughly comparable and
similar-appearing complexity at every level from the view of a
satellite to a microscopic examination of how the water laps up
against individual grains of sand. The branching of trees, which
enables children to use small twigs as stand-ins for full trees in
dioramas, is another
example.

This similarity to naturally occurring phenomena
provides fractals with an everyday familiarity not typically seen
with mathematically generated functions. As a consequence, they can
produce strikingly beautiful results such as the Mandelbrot
set. Intriguingly, fractals have also found a place in CG, or
computer-generated movie effects, where their ability to create
very complex curves with fractal symmetries results in more
realistic virtual
worlds.

### Symmetry combinations

## Symmetry in science

### Symmetry in physics

Symmetry in physics has been generalized to mean
invariance—that is,
lack of any visible change—under any kind of transformation. 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, as a gross oversimplification, states that for
every continuous mathematical symmetry, there is a corresponding
conserved quantity; 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.

### Symmetry in physical objects

#### Classical objects

Although an everyday object may appear exactly
the same after a symmetry operation such as a rotation or an
exchange of two identical parts has been performed on it, it is
readily apparent that such a symmetry is true only as an
approximation for any ordinary physical object.

For example, if one rotates a precisely machined
aluminum equilateral
triangle 120 degrees around its center, a casual observer
brought in before and after the rotation will be unable to decide
whether or not such a rotation took place. However, the reality is
that each corner of a triangle will always appear unique when
examined with sufficient precision. An observer armed with
sufficiently detailed measuring equipment such as optical
or electron
microscopes will not be fooled; he will immediately recognize
that the object has been rotated by looking for details such as
crystals or minor
deformities.

Such simple thought
experiments show that assertions of symmetry in everyday
physical objects are always a matter of approximate similarity
rather than of precise mathematical sameness. The most important
consequence of this approximate nature of symmetries in everyday
physical objects is that such symmetries have minimal or no impacts
on the physics of such objects. Consequently, only the deeper
symmetries of space and time play a major role in classical
physics—that is, the physics of large, everyday objects.

#### Quantum objects

Remarkably, there exists a realm of physics for
which mathematical assertions of simple symmetries in real objects
cease to be approximations. That is the domain of quantum
physics, which for the most part is the physics of very small,
very simple objects such as electrons, protons, light, and atoms.

Unlike everyday objects, objects such as electrons have very limited
numbers of configurations, called states,
in which they can exist. This means that when symmetry operations
such as exchanging the positions of components are applied to them,
the resulting new configurations often cannot be distinguished from
the originals no matter how diligent an observer is. Consequently,
for sufficiently small and simple objects the generic mathematical
symmetry assertion F(x) = x ceases to be approximate, and instead
becomes an experimentally precise and accurate description of the
situation in the real world.

#### Consequences of quantum symmetry

While it makes sense that symmetries could become
exact when applied to very simple objects, the immediate intuition
is that such a detail should not affect the physics of such objects
in any significant way. This is in part because it is very
difficult to view the concept of exact similarity as physically
meaningful. Our mental picture of such situations is invariably the
same one we use for large objects: We picture objects or
configurations that are very, very similar, but for which if we
could "look closer" we would still be able to tell the
difference.

However, the assumption that exact symmetries in
very small objects should not make any difference in their physics
was discovered in the early 1900s to be spectacularly incorrect.
The situation was succinctly summarized by Richard
Feynman in the direct transcripts of his
Feynman Lectures on Physics, Volume III, Section 3.4, Identical
particles. (Unfortunately, the quote was edited out of the printed
version of the same lecture.)

- "... if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails."

The word "interferes"
in this context is a quick way of saying that such objects fall
under the rules of quantum
mechanics, in which they behave more like waves that interfere than like
everyday large objects.

In short, when an object becomes so simple that a
symmetry assertion of the form F(x) = x becomes an exact statement
of experimentally verifiable sameness, x ceases to follow the rules
of classical
physics and must instead be modeled using the more complex—and
often far less intuitive—rules of quantum
physics.

This transition also provides an important
insight into why the mathematics of symmetry are so deeply
intertwined with those of quantum mechanics. When physical systems
make the transition from symmetries that are approximate to ones
that are exact, the mathematical expressions of those symmetries
cease to be approximations and are transformed into precise
definitions of the underlying nature of the objects. From that
point on, the correlation of such objects to their mathematical
descriptions becomes so close that it is difficult to separate the
two.

### Symmetry as a unifying principle of geometry

The German geometer Felix Klein
enunciated a very influential Erlangen
programme in 1872, suggesting symmetry as unifying and
organising principle in geometry (at a time when that was read
'geometries'). This is a broad rather than deep principle.
Initially it led to interest in the groups
attached to geometries, and the slogan transformation
geometry (an aspect of the New Math, but
hardly controversial in modern mathematical practice). By now it
has been applied in numerous forms, as kind of standard attack on
problems.

### Symmetry in mathematics

An example of a mathematical expression
exhibiting symmetry is a²c + 3ab + b²c. If a and b are exchanged,
the expression remains unchanged due to the commutativity of addition
and multiplication.

Like in geometry, for the terms there are two
possibilities:

- It is itself symmetric
- It has one or more other terms symmetric with it, in accordance with the symmetry kind

See also symmetric
function, duality
(mathematics)

### Symmetry in logic

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"
(∧, \land, or &), "or"
(∨), "biconditional"
(iff) (↔), NAND
("not-and"), XOR
("not-biconditional"), and NOR
("not-or").

### Generalizations of symmetry

If we have a given set of objects with some
structure, then it is possible for a symmetry to merely convert
only one object into another, instead of acting upon all possible
objects simultaneously. This requires a generalization from the
concept of symmetry
group to that of a groupoid. Indeed, A. Connes in
his book `Non-commutative_geometry'
writes that Heisenberg discovered quantum mechanics by considering
the groupoid of transitions of the hydrogen spectrum.

The notion of groupoid also leads to notions of
multiple groupoids, namely sets with many compatible groupoid
structures, a structure which trivialises to abelian groups if one
restricts to groups. This leads to prospects of `higher order
symmetry' which have been a little explored, as follows.

The automorphisms of a set, or a set with some
structure, form a group, which models a homotopy 1-type. The
automorphisms of a group G naturally form a crossed
module $G \to Aut(G)$, and crossed modules give an algebraic
model of homotopy 2-types. At the next stage, automorphisms of a
crossed module fit into a structure known as a crossed square, and
this structure is know to give an algebraic model of homotopy
3-types. It is not known how this procedure of generalising
symmetry may be continued, although crossed n-cubes have been
defined and used in algebraic topology, and these structures are
only slowly being brought into theoretical physics. The web site
n-category cafe
has much discussion of n-groups. More information is on `Higher dimensional
group theory'.

Physicists have come up with other directions of
generalization, such as supersymmetry and quantum
groups.

### Symmetry in biology

See symmetry
(biology) and facial
symmetry.

### Symmetry in chemistry

Symmetry is important to chemistry because it explains observations in spectroscopy, quantum chemistry and crystallography. It draws heavily on group theory.## Symmetry in history, religion, and culture

In any human endeavor for which an impressive
visual result is part of the desired objective, symmetries play a
profound role. The innate appeal of symmetry can be found in our
reactions to happening across highly symmetrical natural objects,
such as precisely formed crystals or beautifully spiraled
seashells. Our first reaction in finding such an object often is to
wonder whether we have found an object created by a fellow human,
followed quickly by surprise that the symmetries that caught out
attention are derived from nature itself. In both reactions we give
away our inclination to view symmetries both as beautiful and, in
some fashion, informative of the world around us.

### Symmetry in religious symbols

Row 2. Islamic, Buddhist, Shinto Row 3.
Sikh,
Baha'i,
Jain ]] The
tendency of people to see purpose in symmetry suggests at least one
reason why symmetries are often an integral part of the symbols of
world religions. Just a few of many examples include the sixfold
rotational
symmetry of Judaism's Star of
David, the twofold
point symmetry of Taoism's Taijitu or
Yin-Yang, the bilateral
symmetry of Christianity's
cross and Sikhism's Khanda,
or the fourfold point symmetry of Jain's ancient (and
peacefully intended) version of the swastika. With its strong
prohibitions against the use of representational images, Islam, and in
particular the Sunni branch of
Islam, has developed some of the most intricate and visually
impressive use of symmetries for decorative uses of any major
religion.

The ancient Taijitu image of
Taoism is a
particularly fascinating use of symmetry around a central point,
combined with black-and-white inversion of color at opposite
distances from that central point. The image, which is often
misunderstood in the Western
world as representing good (white) versus evil (black), is
actually intended as a graphical representative of the
complementary need for two abstract concepts of "maleness" (white)
and "femaleness" (black). The symmetry of the symbol in this case
is used not just to create a symbol that catches the attention of
the eye, but to make a significant statement about the
philosophical beliefs of the people and groups that use it. Also an
important symmetrical religious symbol is the Shintoist "Torii"
"The gate of the birds", usually the gate of the Shintoist temples
called "Jinjas".

### Symmetry 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, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you". Peer relationships are based on symmetry, power relationships are based on asymmetry.### Symmetry in architecture

Another human endeavor in which the visual result
plays a vital part in the overall result is architecture. Both in
ancient times, the ability of a large structure to impress or even
intimidate its viewers has often been a major part of its purpose,
and the use of symmetry is an inescapable aspect of how to
accomplish such goals.

Just a few examples of ancient examples of
architectures that made powerful use of symmetry to impress those
around them included the Egyptian Pyramids, the
Greek
Parthenon, and
the first and second Temple
of Jerusalem, China's Forbidden
City, Cambodia's
Angkor
Wat complex, and the many temples and pyramids of ancient
Pre-Columbian
civilizations. More recent historical examples of architectures
emphasizing symmetries include Gothic
architecture cathedrals, and American
President Thomas
Jefferson's Monticello home.
India's
unparalleled Taj Mahal is in
a category by itself, as it may arguably be one of the most
impressive and beautiful uses of symmetry in architecture that the
world has ever seen.

An interesting example of a broken
symmetry in architecture is the Leaning
Tower of Pisa, whose notoriety stems in no small part not for
the intended symmetry of its design, but for the violation of that
symmetry from the lean that developed while it was still under
construction. Modern examples of architectures that make impressive
or complex use of various symmetries include Australia's
astonishing Sydney
Opera House and Houston,
Texas's simpler Astrodome.

Symmetry finds its ways into architecture at
every scale, from the overall external views, through the layout of
the individual floor plans,
and down to the design of individual building elements such as
intricately caved doors, stained
glass windows, tile
mosaics, friezes,
stairwells, stair rails, and balustradess. For sheer
complexity and sophistication in the exploitation of symmetry as an
architectural element, Islamic buildings
such as the Taj Mahal often eclipse those of other cultures and
ages, due in part to the general prohibition of Islam against using
images or people or animals.

Links related to symmetry in architecture
include:

### Symmetry 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. As a minimum, pottery created using a
wheel necessarily begins with full rotational symmetry in its
cross-section, while allowing substantial freedom of shape in the
vertical direction. Upon this inherently symmetrical starting point
cultures from ancient times have tended to add further patterns
that tend to exploit or in many cases reduce the original full
rotational symmetry to a point where some specific visual objective
is achieved. For example, Persian pottery
dating from the fourth millennium B.C. and earlier used symmetric
zigzags, squares, cross-hatchings, and repetitions of figures to
produce more complex and visually striking overall designs.

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 B.C. Bronze vessels exhibited both a
bilateral main motif and a repetitive translated border
design.

Links:

### Symmetry in quilts

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.

Links:

### Symmetry 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 typically
use quadrilateral symmetry—that is, motifs
that are reflected across both the horizontal and vertical
axes.

Links:

### Symmetry in music

Symmetry is of course 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 form
(ABCBA) used by Steve Reich,
Béla
Bartók, and James Tenney
(or swell). In classical music, Bach used the symmetry concepts of
permutation and invariance; see (external link "Fugue No. 21,"
pdf or
Shockwave).

#### Pitch structures

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.

Perle (1992) 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). (Perle, 1990)

#### Equivalency

Tone rows or
pitch
class sets which are
invariant under
retrograde
are horizontally symmetrical, under inversion
vertically. See also Asymmetric
rhythm.

### Symmetry in other arts and crafts

The concept of symmetry is applied to the design
of objects of all shapes and sizes. Other examples include beadwork, furniture, sand
paintings, knotwork,
masks, musical
instruments, and many other endeavors.

### Symmetry in aesthetics

The relationship of symmetry to aesthetics is complex. Certain simple symmetries, and in particular bilateral symmetry, seem to be deeply ingrained in the inherent perception by humans of the likely health or fitness of other living creatures, as can be seen by the simple experiment of distorting one side of the image of an attractive face and asking viewers to rate the attractiveness of the resulting image. Consequently, such symmetries that mimic biology tend to have an innate appeal that in turn drives a powerful tendency to create artifacts with similar symmetry. One only needs to imagine the difficulty in trying to market a highly asymmetrical car or truck to general automotive buyers to understand the power of biologically inspired symmetries such as bilateral symmetry.Another more subtle appeal of symmetry is that of
simplicity, which in turn has an implication of safety, security,
and familiarity. A highly symmetrical room, for example, is
unavoidably also a room in which anything out of place or
potentially threatening can be identified easily and quickly. For
example, people who have grown up in houses full of exact right
angles and precisely identical artifacts can find their first
experience in staying in a room with no exact right angles and no
exactly identical artifacts to be highly disquieting. Symmetry thus
can be a source of comfort not only as an indicator of biological
health, but also of a safe and well-understood living
environment.

Opposed to this is the tendency for excessive
symmetry to be perceived as boring or uninteresting. Humans in
particular have a powerful desire to exploit new opportunities or
explore new possibilities, and an excessive degree of symmetry can
convey a lack of such opportunities.

Yet another possibility is that when symmetries
become too complex or too challenging, the human mind has a
tendency to "tune them out" and perceive them in yet another
fashion: as noise that
conveys no useful information.

Finally, perceptions and appreciation of
symmetries are also dependent on cultural background. The far
greater use of complex geometric symmetries in many Islamic cultures, for
example, makes it more likely that people from such cultures will
appreciate such art forms (or, conversely, to rebel against
them).

As in many human endeavors, the result of the
confluence of many such factors is that effective use of symmetry
in art and architecture is complex, intuitive, and highly dependent
on the skills of the individuals who must weave and combine such
factors within their own creative work. Along with texture, color,
proportion, and other factors, symmetry is a powerful ingredient in
any such synthesis; one only need to examine the Taj Mahal to
powerful role that symmetry plays in determining the aesthetic
appeal of an object.

A few examples of the more explicit use of
symmetries in art can be found in the remarkable art of M. C.
Escher, the creative design of the mathematical concept of a
wallpaper
group, and the many applications (both mathematical and real
world) of tiling.

### Symmetry in games and puzzles

- See also symmetric games.

- See sudoku.

Board games

### Symmetry in literature

See palindrome.

### Moral symmetry

## See also

- Symmetry group
- Chirality
- Fixed points of isometry groups in Euclidean space - center of symmetry
- Spontaneous symmetry breaking
- Gödel, Escher, Bach
- M. C. Escher
- Wallpaper group
- Asymmetry
- Asymmetric rhythm
- Even and odd functions
- Symmetries of polyominoes
- Symmetries of polyiamonds
- Burnside's lemma
- Symmetry (biology)
- Spacetime symmetries
- Semimetric, which is sometimes translated as symmetric in Russian texts.

## References

- Livio, Mario (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. New York: Simon & Schuster. ISBN 0-7432-5820-7.
- Perle, George (1990). The Listening Composer, p. 112. California: University of California Press. ISBN 0-520-06991-9.
- Perle, George (1992). Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 81-96.
- Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
- Weyl, Hermann (1952). Symmetry. Princeton University Press. ISBN 0-691-02374-3.
- Hahn, Werner (1998). Symmetry As A Developmental Principle In Nature And Art World Scientific. ISBN 981-02-2363-3.
- Symmetry: Culture and Science, published by Symmetrion, Budapest. ISSN 0865-4824.
- Darvas, György (2007). Symmetry, Basel-Berlin-Boston: Birkhäuser Verlag, xi + 508 p.
- Petitjean, Michel (2003). Chirality and Symmetry Measures: A Transdisciplinary Review. Entropy 5(3), pp. 271-312.

## Notes

## External links

- An Analysis of the first movement of the Fourth String Quartet (1928) by Andrew Kuster
- Skaalid: Design Theory
- Mathforum: Symmetry/Tesselations
- Calotta: A World of Symmetry
- Dutch: Symmetry Around a Point in the Plane
- Saw: Design Notes
- Chapman: Aesthetics of Symmetry
- Abas: The Wonder Of Symmetry
- ISIS Symmetry
- Symmetry and Asymmetry at The Dictionary of the History of Ideas
- Examples of asymmetry in musical waveforms
- International Symmetry Asociation - ISA
- Institute Symmetrion
- Professor Ian Stewart on the history of symmetry
- Photographs of Symmetrical Cloisters

symmetry in Arabic: تناظر

symmetry in Catalan: Simetria

symmetry in Czech: Symetrie

symmetry in Danish: Symmetri

symmetry in German: Symmetrie (Geometrie)

symmetry in Spanish: Simetría

symmetry in Esperanto: Simetrio

symmetry in French: Symétrie

symmetry in Ido: Simetreso

symmetry in Indonesian: Simetri

symmetry in Hebrew: סימטריה

symmetry in Hungarian: Szimmetria

symmetry in Malay (macrolanguage): Simetri

symmetry in Dutch: Symmetrie

symmetry in Japanese: 対称性

symmetry in Norwegian: Symmetri

symmetry in Polish: Symetria

symmetry in Portuguese: Simetria

symmetry in Russian: Симметрия

symmetry in Simple English: Symmetry

symmetry in Slovenian: Simetrija

symmetry in Serbian: Симетрија

symmetry in Finnish: Symmetria

symmetry in Swedish: Symmetri

symmetry in Turkish: Simetri

symmetry in Ukrainian: Симетрія

symmetry in Chinese: 對稱

# Synonyms, Antonyms and Related Words

accord,
accordance, affinity, agreement, arrangement, array, assent, balance, beauty, chorus, coequality, coextension, coherence, coincidence, communion, community, compatibility, concert, concinnity, concord, concordance, conformance, conformation, conformity, congeniality, congruence, congruency, congruity, congruousness, consistency, consonance, consort, cooperation, corelation, correlation, correlativism, correlativity, correspondence, deployment, disposal, disposition, equality, equation, equilibrium, equipoise, equipollence, equiponderance, equity, equivalence, equivalency, euphony, evenness, finish, formation, harmony, identity, intersection, inverse
proportion, inverse ratio, inverse relationship, justice, layout, levelness, likeness, lineup, marshaling, measure, measuredness, mutuality, oneness, order, orderedness, orderliness, organization, overlap, par, parallelism, parity, peace, poise, proportion, proportionality,
quiet, quietude, rapport, reciprocality, reciprocation, reciprocity, regularity, relativity, rhythm, routine, self-consistency,
setup, structure, sweetness, sync, synchronism, system, tally, timing, tranquillity, uniformity, union, unison, unisonance