# Symmetry

symmetricalsymmetricsymmetriesasymmetricsymmetricallyasymmetricalsymmetry transformationasymmetricallyasymmetrysymmetric relation
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.wikipedia
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### Asymmetry

asymmetricalasymmetricasymmetries
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection).

### Reflection symmetry

reflectionplane of symmetryreflectional symmetry
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.

### Rotational symmetry

axis of symmetryaxisymmetricaxis
Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry, which suits them because food or threats may arrive from any direction.
Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.

### Translational symmetry

translation invarianttranslational invariancetranslation invariance
Discrete translational symmetry is invariant under discrete translation.

### Group (mathematics)

groupgroupsgroup operation
The set of operations that preserve a given property of the object form a group.
Groups share a fundamental kinship with the notion of symmetry.

### Even and odd functions

even functionodd functioneven
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory.
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

### Geometry

geometricgeometricalgeometries
This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries).

### Invariant (mathematics)

invariantinvariantsinvariance
In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling.

### Local symmetry

internal symmetryinternal symmetrieslocal
Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime; internal symmetries of particles; and supersymmetry of physical theories.
In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold.

### Abstract algebra

algebraalgebraicmodern algebra
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory.

### Molecular symmetry

symmetrypoint grouporbital symmetry
The control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects.
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry.

### Discrete symmetry

discrete symmetries
Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime; internal symmetries of particles; and supersymmetry of physical theories.
In mathematics and theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group—e.g. a topological group with a discrete topology whose elements form a finite or a countable set.

### Rotation (mathematics)

rotationrotationsrotate
In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling.
Rotations define important classes of symmetry: rotational symmetry is an invariance with respect to a particular rotation.

### Group theory

groupgroup theoreticalgroup-theoretic
The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory.
Finite groups often occur when considering symmetry of mathematical or

### General covariance

diffeomorphism invariancegenerally covariantcovariant
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations.
A more modern interpretation of the physical content of the original principle of general covariance is that the Lie group GL 4 (R) is a fundamental "external" symmetry of the world.

### Symmetric matrix

symmetricsymmetric matricessymmetrical
Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois groups in Galois theory.
Other types of symmetry or pattern in square matrices have special names; see for example:

### Arch form

archarch (swell) form
Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney.
In 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.

### Noether's theorem

Noether currentNoether chargeNoether theorem
See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity such as energy or momentum; a conserved current, in Noether's original language); and also, Wigner's classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.
Then, we say Q is a generator of a one parameter symmetry Lie group.

### Rectangle

rectangularoblongrectangles
Not surprisingly, rectangular rugs have typically the symmetries of a rectangle—that is, motifs that are reflected across both the horizontal and vertical axes (see ).
Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects.

### M. C. Escher

M.C. EscherEscherMaurits Cornelis Escher
Symmetries are central to the art of M.C. Escher and the many applications of tessellation in art and craft forms such as wallpaper, ceramic tilework such as in Islamic geometric decoration, batik, ikat, carpet-making, and many kinds of textile and embroidery patterns.
His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations.

### Symmetry in biology

Humans find bilateral symmetry in faces physically attractive; it indicates health and genetic fitness.
In nature and biology, symmetry is always approximate.

### Sandpainting

sand paintingNavajo sandpaintingbonsedi
The order and symmetry of the painting symbolise the harmony which a patient wishes to reestablish in his or her life.

### Automorphism

field automorphismautomorphismsAut
It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure.

### Burnside's lemma

Cauchy–Frobenius lemmaorbit counting lemmaPólya–Burnside lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, orbit-counting theorem, or The Lemma that is not Burnside's, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects.

### Spacetime symmetries

Symmetries in General Relativitysymmetries4-dimensional spacetimes
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry.