Symmetry
Overview

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
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

: by geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, through physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

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.
Encyclopedia
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
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

: by geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, through physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

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
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

;
• as a spatial relationship
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

;
• through geometric transformations such as scaling
Scaling (geometry)
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...

, reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

, and rotation
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...

;
• through other kinds of functional transformations; and
• as an aspect of abstract object
Abstract object
An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...

s, theoretic models, language
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

, music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

and even knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

itself.

This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

as a whole. The third perspective describes symmetry as it relates to science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

and technology
Technology
Technology is the making, usage, and knowledge of tools, machines, techniques, crafts, systems or methods of organization in order to solve a problem or perform a specific function. It can also refer to the collection of such tools, machinery, and procedures. The word technology comes ;...

. In this context, symmetries underlie some of the most profound results found in modern physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, including aspects of space and time
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. Finally, a fourth perspective discusses symmetry in the humanities
Humanities
The humanities are academic disciplines that study the human condition, using methods that are primarily analytical, critical, or speculative, as distinguished from the mainly empirical approaches of the natural sciences....

, covering its rich and varied use in history
History
History is the discovery, collection, organization, and presentation of information about past events. History can also mean the period of time after writing was invented. Scholars who write about history are called historians...

, architecture
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...

, art
Art
Art is the product or process of deliberately arranging items in a way that influences and affects one or more of the senses, emotions, and intellect....

, and religion
Religion
Religion is a collection of cultural systems, belief systems, and worldviews that establishes symbols that relate humanity to spirituality and, sometimes, to moral values. Many religions have narratives, symbols, traditions and sacred histories that are intended to give meaning to life or to...

.

The opposite of symmetry is asymmetry
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....

.

## Symmetry in geometry

The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...

of isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.

### 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
Mirror image
A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...

).

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
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

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" or "T has left-right symmetry."

The triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s with this symmetry are isosceles, the quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

s with this symmetry are the kites
Kite (geometry)
In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

and the isosceles trapezoid
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s.

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...

s 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
Bilateria
The bilateria are all animals having a bilateral symmetry, i.e. they have a front and a back end, as well as an upside and downside. Radially symmetrical animals like jellyfish have a topside and downside, but no front and back...

(bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane
Sagittal plane
Sagittal plane is a vertical plane which passes from front to rear dividing the body into right and left sections.-Variations:Examples include:...

.

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
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

).

For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:
• with respect to a non-isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

affine
Affine
Affine may refer to:*Affine cipher, a special case of the more general substitution cipher*Affine combination, a certain kind of constrained linear combination*Affine connection, a connection on the tangent bundle of a differentiable manifold...

involution
Point reflection
In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space...

(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
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m).

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
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

with determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

1. For m=3 this is the rotation group
Rotation group
In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...

.

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
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also rotational invariance
Rotational invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument...

.

### Translational symmetry

Translational symmetry leaves an object invariant under a discrete or continuous group of translations
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

.

### Glide reflection symmetry

A glide reflection
Glide reflection
In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result...

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
Improper rotation
In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...

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 group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

s, 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
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...

.

### Helical symmetry

Helical
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

symmetry is the kind of symmetry seen in such everyday objects as springs
Spring (device)
A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...

Slinky or "Lazy Spring" is a toy consisting of a helical spring that stretches and can bounce up and down. It can perform a number of tricks, including traveling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its own momentum.-History:The toy was...

toys, drill bits, and auger
Auger
An auger is a drilling device, or drill bit, that usually includes a rotating helical screw blade called a "flighting" to act as a screw conveyor to remove the drilled out material...

s. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis
Screw axis
The screw axis of an object is a line that is simultaneously the axis of rotation and the line along which a translation occurs...

. 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 is 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
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

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
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

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
Spring (device)
A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...

Slinky or "Lazy Spring" is a toy consisting of a helical spring that stretches and can bounce up and down. It can perform a number of tricks, including traveling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its own momentum.-History:The toy was...

, drill bit
Drill bit
Drill bits are cutting tools used to create cylindrical holes. Bits are held in a tool called a drill, which rotates them and provides torque and axial force to create the hole. Specialized bits are also available for non-cylindrical-shaped holes....

s, and auger
Auger
An auger is a drilling device, or drill bit, that usually includes a rotating helical screw blade called a "flighting" to act as a screw conveyor to remove the drilled out material...

s. 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 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°/, see e.g. double helix. This concept can be further generalized to include cases where 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 required to observe the symmetry is irrational. The angle of rotation 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
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

is an example of this type of non-repeating helical symmetry.

### Non-isometric symmetries

A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
• The group of similarity transformations, i.e. affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

s represented by a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

A that is a scalar times an orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

. Thus dilations
Dilation (mathematics)
In mathematics, a dilation is a function f from a metric space into itself that satisfies the identityd=rd \,for all points where d is the distance from x to y and r is some positive real number....

Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

is considered a symmetry.

• The group of affine transformations represented by a matrix A with determinant 1 or −1, i.e. the transformations which preserve area; this adds e.g. oblique reflection symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

.

• The group of all bijective affine transformations.

• The group of Möbius transformations which preserve cross-ratio
Cross-ratio
In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...

s.

In Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

's Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, whereas the group of Möbius transformations defines projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

.

### 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 elephant
Elephant
Elephants are large land mammals in two extant genera of the family Elephantidae: Elephas and Loxodonta, with the third genus Mammuthus extinct...

s versus mice
Mouse
A mouse is a small mammal belonging to the order of rodents. The best known mouse species is the common house mouse . It is also a popular pet. In some places, certain kinds of field mice are also common. This rodent is eaten by large birds such as hawks and eagles...

, 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 fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s. As conceived by Benoît Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

, fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of magnification
Magnification
Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

is used to examine it. A coast
Coast
A coastline or seashore is the area where land meets the sea or ocean. A precise line that can be called a coastline cannot be determined due to the dynamic nature of tides. The term "coastal zone" can be used instead, which is a spatial zone where interaction of the sea and land processes occurs...

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 diorama
Diorama
The word diorama can either refer to a nineteenth century mobile theatre device, or, in modern usage, a three-dimensional full-size or miniature model, sometimes enclosed in a glass showcase for a museum...

s, 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
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...

. 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 world
Virtual world
A virtual world is an online community that takes the form of a computer-based simulated environment through which users can interact with one another and use and create objects. The term has become largely synonymous with interactive 3D virtual environments, where the users take the form of...

s.

## Symmetry in mathematics

In formal terms, we say that a mathematical object
Mathematical object
In mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...

is symmetric with respect to a given mathematical operation
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic 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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

.
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).

### Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

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 field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s, 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
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

. 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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, G is the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...

E(n), the group of isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

, 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

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
Fundamental domain
In geometry, the fundamental domain of a symmetry group of an object is a part or pattern, as small or irredundant as possible, which determines the whole object based on the symmetry. More rigorously, given a topological space and a group acting on it, the images of a single point under the group...

. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....

.

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
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

with a current perpendicular to the axis; the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

(a pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

) 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
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

and velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

, 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 {...,1,2,5,6,9,10,13,14,...} acts transitively on all these points, while {...,1,2,3,5,6,7,9,10,11,13,14,15,...} does not act transitively on all points. Equivalently, the first set is only one conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...

with respect to isometries, while the second has two classes.

### Symmetric functions

A symmetric function is a function which is unchanged by any permutation of its variables. For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2 – yz is not.

A function may be unchanged by a sub-group of all the permutations of its variables. For example, ac + 3ab + bc is unchanged if a and b are exchanged; its symmetry group is isomorphic to C2.

### Symmetry in logic

Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

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 connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s are "and
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

" (∧, , or &), "or
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

" (∨), "biconditional
Logical biconditional
In logic and mathematics, the logical biconditional is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis and p is a conclusion...

" (if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR
Logical NOR
In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form is true precisely when neither p nor q is true—i.e. when both of p and q are false...

("not-or").

### Symmetry in physics

Symmetry in physics has been generalized to mean invariance
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...

—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable 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
Philip Warren Anderson
Philip Warren Anderson is an American physicist and Nobel laureate. Anderson has made contributions to the theories of localization, antiferromagnetism and high-temperature superconductivity.- Biography :...

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
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

(which, in greatly simplified form, 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
Wigner's classification
In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues...

, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

#### 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
Optical microscope
The optical microscope, often referred to as the "light microscope", is a type of microscope which uses visible light and a system of lenses to magnify images of small samples. Optical microscopes are the oldest design of microscope and were possibly designed in their present compound form in the...

or electron microscope
Electron microscope
An electron microscope is a type of microscope that uses a beam of electrons to illuminate the specimen and produce a magnified image. Electron microscopes have a greater resolving power than a light-powered optical microscope, because electrons have wavelengths about 100,000 times shorter than...

s will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

s or minor deformities.

Such simple thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

s 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
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

—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 electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s, light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

, and atoms.

Unlike everyday objects, objects such as electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s 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
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...

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
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, in which they behave more like wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s 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
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

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.

### 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
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

to that of a groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...

. 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
Crossed module
In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H , and a homomorphism of groups...

, 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 known 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.

Physicists have come up with other directions of generalization, such as supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...

and quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

s, yet the different options are indistinguishable during various circumstances.

### Symmetry in chemistry

Symmetry is important to chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

because it explains observations in spectroscopy
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

, quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

and crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

. It draws heavily on group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

.

## 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 our 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

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
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

of Judaism
Judaism
Judaism ) is the "religion, philosophy, and way of life" of the Jewish people...

's Star of David
Star of David
The Star of David, known in Hebrew as the Shield of David or Magen David is a generally recognized symbol of Jewish identity and Judaism.Its shape is that of a hexagram, the compound of two equilateral triangles...

, the twofold point symmetry of Taoism
Taoism
Taoism refers to a philosophical or religious tradition in which the basic concept is to establish harmony with the Tao , which is the mechanism of everything that exists...

's Taijitu
Taijitu
Taijitu is a term which refers to a Chinese symbol for the concept of yin and yang...

or Yin-Yang, the bilateral symmetry of Christianity
Christianity
Christianity is a monotheistic religion based on the life and teachings of Jesus as presented in canonical gospels and other New Testament writings...

's cross
Cross
A cross is a geometrical figure consisting of two lines or bars perpendicular to each other, dividing one or two of the lines in half. The lines usually run vertically and horizontally; if they run obliquely, the design is technically termed a saltire, although the arms of a saltire need not meet...

and Sikhism
Sikhism
Sikhism is a monotheistic religion founded during the 15th century in the Punjab region, by Guru Nanak Dev and continued to progress with ten successive Sikh Gurus . It is the fifth-largest organized religion in the world and one of the fastest-growing...

's Khanda, or the fourfold point symmetry of Hindu
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...

's ancient version of the swastika
Swastika
The swastika is an equilateral cross with its arms bent at right angles, in either right-facing form in counter clock motion or its mirrored left-facing form in clock motion. Earliest archaeological evidence of swastika-shaped ornaments dates back to the Indus Valley Civilization of Ancient...

. With its strong prohibitions against the use of representational images, Islam
Islam
Islam . The most common are and .   : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...

, and in particular the Sunni branch of Islam, has developed intricate use of symmetries.

### 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
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...

. 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 Egypt
Egypt
Egypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...

ian Pyramids, the Greek
Greece
Greece , officially the Hellenic Republic , and historically Hellas or the Republic of Greece in English, is a country in southeastern Europe....

Parthenon
Parthenon
The Parthenon is a temple on the Athenian Acropolis, Greece, dedicated to the Greek goddess Athena, whom the people of Athens considered their virgin patron. Its construction began in 447 BC when the Athenian Empire was at the height of its power. It was completed in 438 BC, although...

, the first and second Temple of Jerusalem, China's Forbidden City
Forbidden City
The Forbidden City was the Chinese imperial palace from the Ming Dynasty to the end of the Qing Dynasty. It is located in the middle of Beijing, China, and now houses the Palace Museum...

, Cambodia
Cambodia
Cambodia , officially known as the Kingdom of Cambodia, is a country located in the southern portion of the Indochina Peninsula in Southeast Asia...

's Angkor Wat
Angkor Wat
Angkor Wat is a temple complex at Angkor, Cambodia, built for the king Suryavarman II in the early 12th century as his state temple and capital city. As the best-preserved temple at the site, it is the only one to have remained a significant religious centre since its foundation – first Hindu,...

complex, and the many temples and pyramids of ancient Pre-Columbian
Pre-Columbian
The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the American continents, spanning the time of the original settlement in the Upper Paleolithic period to European colonization during...

civilizations. More recent historical examples of architectures emphasizing symmetries include Gothic architecture
Gothic architecture
Gothic architecture is a style of architecture that flourished during the high and late medieval period. It evolved from Romanesque architecture and was succeeded by Renaissance architecture....

cathedrals, and American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

President Thomas Jefferson
Thomas Jefferson
Thomas Jefferson was the principal author of the United States Declaration of Independence and the Statute of Virginia for Religious Freedom , the third President of the United States and founder of the University of Virginia...

's Monticello
Monticello
Monticello is a National Historic Landmark just outside Charlottesville, Virginia, United States. It was the estate of Thomas Jefferson, the principal author of the United States Declaration of Independence, third President of the United States, and founder of the University of Virginia; it is...

home. The Taj Mahal
Taj Mahal
The Taj Mahal is a white Marble mausoleum located in Agra, India. It was built by Mughal emperor Shah Jahan in memory of his third wife, Mumtaz Mahal...

is also an example of symmetry.

An interesting example of a broken symmetry in architecture is the Leaning Tower of Pisa
Leaning Tower of Pisa
The Leaning Tower of Pisa or simply the Tower of Pisa is the campanile, or freestanding bell tower, of the cathedral of the Italian city 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
Australia
Australia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...

's Sydney Opera House
Sydney Opera House
The Sydney Opera House is a multi-venue performing arts centre in the Australian city of Sydney. It was conceived and largely built by Danish architect Jørn Utzon, finally opening in 1973 after a long gestation starting with his competition-winning design in 1957...

and Houston, Texas
Houston, Texas
Houston is the fourth-largest city in the United States, and the largest city in the state of Texas. According to the 2010 U.S. Census, the city had a population of 2.1 million people within an area of . Houston is the seat of Harris County and the economic center of , which is the ...

's simpler Astrodome.

Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual floor plan
Floor plan
In architecture and building engineering, a floor plan, or floorplan, is a diagram, usually to scale, showing a view from above of the relationships between rooms, spaces and other physical features at one level of a structure....

s, and down to the design of individual building elements such as intricately caved doors, stained glass windows, tile mosaics
Mosaic
Mosaic is the art of creating images with an assemblage of small pieces of colored glass, stone, or other materials. It may be a technique of decorative art, an aspect of interior decoration, or of cultural and spiritual significance as in a cathedral...

, frieze
Frieze
thumb|267px|Frieze of the [[Tower of the Winds]], AthensIn architecture the frieze is the wide central section part of an entablature and may be plain in the Ionic or Doric order, or decorated with bas-reliefs. Even when neither columns nor pilasters are expressed, on an astylar wall it lies upon...

s, stairwells, stair rails, and balustradess. For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, Islam
Islam
Islam . The most common are and .   : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...

ic 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 of people or animals.

### 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
Chinese people
The term Chinese people may refer to any of the following:*People with Han Chinese ethnicity ....

, 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.

### Symmetry in quilts

As quilt
Quilt
A quilt is a type of bed cover, traditionally composed of three layers of fiber: a woven cloth top, a layer of batting or wadding and a woven back, combined using the technique of quilting. “Quilting” refers to the technique of joining at least two fabric layers by stitches or ties...

s 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.

### Symmetry in carpets and rugs

A long tradition of the use of symmetry in carpet
Carpet
A carpet is a textile floor covering consisting of an upper layer of "pile" attached to a backing. The pile is generally either made from wool or a manmade fibre such as polypropylene,nylon or polyester and usually consists of twisted tufts which are often heat-treated to maintain their...

and rug patterns spans a variety of cultures. American Navajo
Navajo people
The Navajo of the Southwestern United States are the largest single federally recognized tribe of the United States of America. The Navajo Nation has 300,048 enrolled tribal members. The Navajo Nation constitutes an independent governmental body which manages the Navajo Indian reservation in the...

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.

### Symmetry in music

File:Major and minor triads.png|300px|thumb|right|Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)

poly 35 442 35 544 179 493 root of A minor triad
A minor
A minor is a minor scale based on A, consisting of the pitches A, B, C, D, E, F, and G. The harmonic minor scale raises the G to G...

poly 479 462 446 493 479 526 513 492 third of A minor triad
A minor
A minor is a minor scale based on A, consisting of the pitches A, B, C, D, E, F, and G. The harmonic minor scale raises the G to G...

poly 841 472 782 493 840 514 821 494 fifth of A minor triad
A minor
A minor is a minor scale based on A, consisting of the pitches A, B, C, D, E, F, and G. The harmonic minor scale raises the G to G...

poly 926 442 875 460 906 493 873 525 926 545 fifth of A minor triad
A minor
A minor is a minor scale based on A, consisting of the pitches A, B, C, D, E, F, and G. The harmonic minor scale raises the G to G...

poly 417 442 417 544 468 525 437 493 469 459 root of C major triad
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....

poly 502 472 522 493 502 514 560 493 root of C major triad
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....

poly 863 462 830 493 863 525 895 493 third of C major triad
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....

poly 1303 442 1160 493 1304 544 fifth of C major triad
C major
C major is a musical major scale based on C, with pitches C, D, E, F, G, A, and B. Its key signature has no flats/sharps.Its relative minor is A minor, and its parallel minor is C minor....

poly 280 406 264 413 282 419 275 413 fifth of E minor triad
E minor
E minor is a minor scale based on the note E. The E natural minor scale consists of the pitches E, F, G, A, B, C, and D. The E harmonic minor scale contains the natural 7, D, rather than the flatted 7, D – to align with the major dominant chord, B7 .Its key signature has one sharp, F .Its...

poly 308 397 293 403 301 412 294 423 309 428 fifth of E minor triad
E minor
E minor is a minor scale based on the note E. The E natural minor scale consists of the pitches E, F, G, A, B, C, and D. The E harmonic minor scale contains the natural 7, D, rather than the flatted 7, D – to align with the major dominant chord, B7 .Its key signature has one sharp, F .Its...

poly 844 397 844 428 886 413 root of E minor triad
E minor
E minor is a minor scale based on the note E. The E natural minor scale consists of the pitches E, F, G, A, B, C, and D. The E harmonic minor scale contains the natural 7, D, rather than the flatted 7, D – to align with the major dominant chord, B7 .Its key signature has one sharp, F .Its...

poly 1240 404 1230 412 1239 422 1250 412 third of E minor triad
E minor
E minor is a minor scale based on the note E. The E natural minor scale consists of the pitches E, F, G, A, B, C, and D. The E harmonic minor scale contains the natural 7, D, rather than the flatted 7, D – to align with the major dominant chord, B7 .Its key signature has one sharp, F .Its...

poly 289 404 279 413 288 422 300 413 third of G major triad
G major
G major is a major scale based on G, with the pitches G, A, B, C, D, E, and F. Its key signature has one sharp, F; in treble-clef key signatures, the sharp-symbol for F is usually placed on the first line from the top, though in some Baroque music it is placed on the first space from the bottom...

poly 689 398 646 413 689 429 fifth of G major triad
G major
G major is a major scale based on G, with the pitches G, A, B, C, D, E, and F. Its key signature has one sharp, F; in treble-clef key signatures, the sharp-symbol for F is usually placed on the first line from the top, though in some Baroque music it is placed on the first space from the bottom...

poly 1221 397 1222 429 1237 423 1228 414 1237 403 root of G major triad
G major
G major is a major scale based on G, with the pitches G, A, B, C, D, E, and F. Its key signature has one sharp, F; in treble-clef key signatures, the sharp-symbol for F is usually placed on the first line from the top, though in some Baroque music it is placed on the first space from the bottom...

poly 1249 406 1254 413 1249 418 1265 413 root of G major triad
G major
G major is a major scale based on G, with the pitches G, A, B, C, D, E, and F. Its key signature has one sharp, F; in treble-clef key signatures, the sharp-symbol for F is usually placed on the first line from the top, though in some Baroque music it is placed on the first space from the bottom...

poly 89 567 73 573 90 579 86 573 fifth of D minor triad
D minor
D minor is a minor scale based on D, consisting of the pitches D, E, F, G, A, B, and C. In the harmonic minor, the C is raised to C. Its key signature has one flat ....

poly 117 558 102 563 111 572 102 583 118 589 fifth of D minor triad
D minor
D minor is a minor scale based on D, consisting of the pitches D, E, F, G, A, B, and C. In the harmonic minor, the C is raised to C. Its key signature has one flat ....

poly 650 558 650 589 693 573 root of D minor triad
D minor
D minor is a minor scale based on D, consisting of the pitches D, E, F, G, A, B, and C. In the harmonic minor, the C is raised to C. Its key signature has one flat ....

poly 1050 563 1040 574 1050 582 1061 574 third of D minor triad
D minor
D minor is a minor scale based on D, consisting of the pitches D, E, F, G, A, B, and C. In the harmonic minor, the C is raised to C. Its key signature has one flat ....

poly 98 565 88 573 98 583 110 574 third of F major triad
F major
F major is a musical major scale based on F, consisting of the pitches F, G, A, B, C, D, and E. Its key signature has one flat . It is by far the oldest key signature with an accidental, predating the others by hundreds of years...

poly 498 558 455 573 498 589 fifth of F major triad
F major
F major is a musical major scale based on F, consisting of the pitches F, G, A, B, C, D, and E. Its key signature has one flat . It is by far the oldest key signature with an accidental, predating the others by hundreds of years...

poly 1031 557 1031 589 1047 583 1038 574 1046 563 root of F major triad
F major
F major is a musical major scale based on F, consisting of the pitches F, G, A, B, C, D, and E. Its key signature has one flat . It is by far the oldest key signature with an accidental, predating the others by hundreds of years...

poly 1075 573 1059 580 1064 573 1058 567 root of F major triad
F major
F major is a musical major scale based on F, consisting of the pitches F, G, A, B, C, D, and E. Its key signature has one flat . It is by far the oldest key signature with an accidental, predating the others by hundreds of years...

desc none

Symmetry is not restricted to the visual arts. Its role in the history of music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

touches many aspects of the creation and perception of music.

#### Musical form

Symmetry has been used as a formal
Musical form
The term musical form refers to the overall structure or plan of a piece of music, and it describes the layout of a composition as divided into sections...

constraint by many composers, such as the arch (swell) form
Arch form
In music, arch form is a sectional structure for a piece of music based on repetition, in reverse order, of all or most musical sections such that the overall form is symmetric, most often around a central movement...

(ABCBA) used by Steve Reich
Steve Reich
Stephen Michael "Steve" Reich is an American composer who together with La Monte Young, Terry Riley, and Philip Glass is a pioneering composer of minimal music...

, Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...

, and James Tenney
James Tenney
James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...

. In classical music, Bach used the symmetry concepts of permutation and invariance.

#### Pitch structures

Symmetry is also an important consideration in the formation of scales and chords
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...

music being made up of non-symmetrical groups of pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

, such as the diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...

or the major chord
Major chord
In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...

. Symmetrical scales or chords, such as the whole tone scale
Whole tone scale
In music, a whole tone scale is a scale in which each note is separated from its neighbors by the interval of a whole step. There are only two complementary whole tone scales, both six-note or hexatonic scales:...

, augmented chord, or diminished seventh chord
Seventh chord
A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a major triad with an added minor seventh...

(diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key
Key (music)
In music theory, the term key is used in many different and sometimes contradictory ways. A common use is to speak of music as being "in" a specific key, such as in the key of C major or in the key of F-sharp. Sometimes the terms "major" or "minor" are appended, as in the key of A minor or in the...

or tonal center, and have a less specific diatonic functionality. However, composers such as Alban Berg
Alban Berg
Alban Maria Johannes Berg was an Austrian composer. He was a member of the Second Viennese School with Arnold Schoenberg and Anton Webern, and produced compositions that combined Mahlerian Romanticism with a personal adaptation of Schoenberg's twelve-tone technique.-Early life:Berg was born in...

, Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...

, and George Perle
George Perle
George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...

have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...

tonal center
Tonic (music)
In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...

s.

Perle (1992) explains "C–E, D–F#, [and] Eb–G, are different instances of the same interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

...the other kind of identity. ..has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"
 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).
 + 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
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions
Chord progression
A chord progression is a series of musical chords, or chord changes that "aims for a definite goal" of establishing a tonality founded on a key, root or tonic chord. In other words, the succession of root relationships...

in the works of Romantic
Romantic music
Romantic music or music in the Romantic Period is a musicological and artistic term referring to a particular period, theory, compositional practice, and canon in Western music history, from 1810 to 1900....

composers such as Gustav Mahler
Gustav Mahler
Gustav Mahler was a late-Romantic Austrian composer and one of the leading conductors of his generation. He was born in the village of Kalischt, Bohemia, in what was then Austria-Hungary, now Kaliště in the Czech Republic...

and Richard Wagner
Richard Wagner
Wilhelm Richard Wagner was a German composer, conductor, theatre director, philosopher, music theorist, poet, essayist and writer primarily known for his operas...

form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin
Alexander Scriabin
Alexander Nikolayevich Scriabin was a Russian composer and pianist who initially developed a lyrical and idiosyncratic tonal language inspired by the music of Frédéric Chopin. Quite independent of the innovations of Arnold Schoenberg, Scriabin developed an increasingly atonal musical system,...

, Edgard Varèse
Edgard Varèse
Edgard Victor Achille Charles Varèse, , whose name was also spelled Edgar Varèse , was an innovative French-born composer who spent the greater part of his career in the United States....

, 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 row
Tone row
In music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

s or pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

sets
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...

Permutation (music)
In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...

are horizontally symmetrical, under inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...

### 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
Beadwork is the art or craft of attaching beads to one another or to cloth, usually by the use of a needle and thread or soft, flexible wire. Most beadwork takes the form of jewelry or other personal adornment, but beads are also used in wall hangings and sculpture.Beadwork techniques are broadly...

, furniture
Furniture
Furniture is the mass noun for the movable objects intended to support various human activities such as seating and sleeping in beds, to hold objects at a convenient height for work using horizontal surfaces above the ground, or to store things...

, sand paintings, knot
Knot
A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object—the "load"...

work, masks, musical instruments, and many other endeavors.

### Symmetry in aesthetics

The relationship of symmetry to aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...

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
Čar
Čar is a village in the municipality of Bujanovac, Serbia. According to the 2002 census, the town has a population of 296 people.-References:...

or truck
Truck
A truck or lorry is a motor vehicle designed to transport cargo. Trucks vary greatly in size, power, and configuration, with the smallest being mechanically similar to an automobile...

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. Most people display a preference for figures that have a certain degree of simplicity and symmetry, but enough complexity to make them interesting.

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
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical 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 Islam
Islam
Islam . The most common are and .   : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...

ic 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
Taj Mahal
The Taj Mahal is a white Marble mausoleum located in Agra, India. It was built by Mughal emperor Shah Jahan in memory of his third wife, Mumtaz Mahal...

to powerful role that symmetry plays in determining the aesthetic appeal of an object.

Modernist architecture rejects symmetry, stating 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. This notion of getting rid of symmetry was first encountered in International style
International style (architecture)
The International style is a major architectural style that emerged in the 1920s and 1930s, the formative decades of Modern architecture. The term originated from the name of a book by Henry-Russell Hitchcock and Philip Johnson, The International Style...

. Some people find asymmetrical layouts of buildings and structures revolutionizing; other find them restless, boring and unnatural.

A few examples of the more explicit use of symmetries in art can be found in the remarkable art of M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

, the creative design of the mathematical concept of a wallpaper group
Wallpaper group
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...

, and the many applications (both mathematical and real world) of tiling
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

.

Symmetry in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

• Skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

, asymmetry of a statistical distribution

Symmetry in games and puzzles
• Symmetric games
• Sudoku
Sudoku
is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contains all of the digits from 1 to 9...

Symmetry in literature
• Palindrome
Palindrome
A palindrome is a word, phrase, number, or other sequence of units that can be read the same way in either direction, with general allowances for adjustments to punctuation and word dividers....

Moral symmetry
• Empathy
Empathy
Empathy is the capacity to recognize and, to some extent, share feelings that are being experienced by another sapient or semi-sapient being. Someone may need to have a certain amount of empathy before they are able to feel compassion. The English word was coined in 1909 by E.B...

& Sympathy
Sympathy
Sympathy is a social affinity in which one person stands with another person, closely understanding his or her feelings. Also known as empathic concern, it is the feeling of compassion or concern for another, the wish to see them better off or happier. Although empathy and sympathy are often used...

• Golden Rule
Golden Rule
Golden Rule may refer to:*The Golden Rule in ethics, morality, history and religion, also known as the ethic of reciprocity*Golden Rule savings rate, in economics, the savings rate which maximizes consumption in the Solow growth model...

• Reciprocity
Reciprocity (social psychology)
Reciprocity in social psychology refers to responding to a positive action with another positive action, rewarding kind actions. People categorize an action as kind by viewing its consequences and also by the person's fundamental intentions. Even if the consequences are the same, underlying...

• Reflective equilibrium
Reflective equilibrium
Reflective equilibrium is a state of balance or coherence among a set of beliefs arrived at by a process of deliberative mutual adjustment among general principles and particular judgments. Although he did not use the term, philosopher Nelson Goodman introduced the method of reflective equilibrium...

• Tit for tat
Tit for tat
Tit for tat is an English saying meaning "equivalent retaliation". It is also a highly effective strategy in game theory for the iterated prisoner's dilemma. It was first introduced by Anatol Rapoport in Robert Axelrod's two tournaments, held around 1980. An agent using this strategy will initially...

Other
• Asymmetric rhythm
• Asymmetry
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....

• Burnside's lemma
Burnside's lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...

• Chirality
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

• M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

• Even and odd functions
Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...

• Fixed points of isometry groups in Euclidean space
Fixed points of isometry groups in Euclidean space
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space....

– center of symmetry
• Gödel, Escher, Bach
Gödel, Escher, Bach
Gödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....

• Ignacio Matte Blanco
Ignacio Matte Blanco
Ignacio Matte Blanco was a Chilean psychiatrist and psychoanalyst who developed a rule-based structure for the unconscious which allows us to make sense of the non-logical aspects of thought...

• Semimetric, which is sometimes translated as symmetric in Russian texts.
• Spacetime symmetries
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...

• Spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

• Symmetric relation
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...

• Symmetries of polyiamonds
• Symmetries of polyominoes
• Symmetry (biology)
Symmetry (biology)
Symmetry in biology is the balanced distribution of duplicate body parts or shapes. The body plans of most multicellular organisms exhibit some form of symmetry, either radial symmetry or bilateral symmetry or "spherical symmetry". A small minority exhibit no symmetry .In nature and biology,...

• Symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

• Time symmetry
T-symmetry
T Symmetry is the symmetry of physical laws under a time reversal transformation: T: t \mapsto -t.Although in restricted contexts one may find this symmetry, the observable universe itself does not show symmetry under time reversal, primarily due to the second law of thermodynamics.Time asymmetries...

• Wallpaper group
Wallpaper group
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...