# 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

**bilateral symmetryradial symmetrybilaterally symmetrical**

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**

Examples include beadwork, furniture, sand paintings, knotwork, masks, and musical instruments.

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.