# introduction of symmetry

Symmetry occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations.

Two objects are symmetric to each other with respect to the invariant transformations if one object is obtained from the other by one of the transformations. It is an equivalence relation.

In the case of symmetric functions, the value of the output is invariant under permutations of variables. These permutations form a group, the symmetric group. In the case of isometric transformations in Euclidean geometry, one uses the term symmetry group. More generally, one uses the term automorphism group.

## Symmetric relations

We call a relation symmetric if every time the relation stands from A to B, it stands too from B to A. Note that symmetry is not the exact opposite of antisymmetry.

More on Symmetric relation.

## Symmetric functions

In the case of symmetric functions, the value of the output is invariant under permutations of variables. From the form of an equation one may observe that certain permutations of the unknowns result in an equivalent equation. In that case the set of solutions is invariant under any permutation of the unknowns in the group generated by the aforementioned permutations. For example

• (a − b)(b − c)(c − a) = 10, for any solution (a,b,c), permutations (a b c) and (a c b) can be applied giving additional solutions (b, c, a) and (c, a, b).
• a2c + 3ab + b2c remains unchanged under interchanging of a and b.
• For a sphere, if φ is the longitude, θ the colatitude, and r the radius, then the great-circle distance is given by Some symmetries clear from the problem can be verified in the formula; the distance is invariant under:

adding the same angle to both longitudes
interchanging longitudes and/or interchanging latitudes
reflecting both colatitudes in the value 90°

## In algebra

A symmetric matrix, seen as a symmetric function of the row- and column number, is an example. The second order partial derivatives of a suitably smooth function, seen as a function of the two indexes, is another example. See also symmetry of second derivatives.

A relation is symmetric if and only if the corresponding boolean-valued function is a symmetric function.

A binary operation is commutative if the operator, as function of two variables, is a symmetric function. Symmetric operators on sets include the union, intersection, and symmetric difference.

The whole subject of Galois theory deals with well-hidden symmetries of fields.

A high-level concept related to symmetry is mathematical duality.

## In geometry

By considering the coordinate space we can consider the symmetry in geometric terms. In the case of three variables we can use e.g. Schoenflies notation for symmetries in 3D. In the example the solution set is geometrically in coordinate space at least of symmetry type C3. If all permutations were allowed this would be C3v. If only two unknowns could be interchanged this would be Cs.

In fact, prior to the 20th century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20th century that the current abstract definition of a group without any reference to group actions was used instead.

## Symmetries of differential equations

A symmetry of a differential equation is a transformation that leaves the differential equation invariant, knowledge of such symmetries may help solve the differential equation.

A Lie symmetry of a system of differential equations is a continuous symmetry of the system of differential equations. Knowledge of a Lie symmetry can be used to simplify an ordinary differential equation through reduction of order.

For ordinary differential equations, knowledge of an appropriate set of Lie symmetries allows one to explicitly calculate a set of first integrals, yielding a complete solution without integration.

Symmetries may be found by solving a related set of ordinary differential equations. Solving these equations is often much simpler than solving the original differential equations.

## Objects symmetric to each other

Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. It is an equivalence relation.

## Randomness

The idea of randomness, without clauses, suggests a probability distribution with "maximum symmetry" with respect to all outcomes.

In the case of finite possible outcomes, symmetry with respect to permutations (relabelings) implies a discrete uniform distribution.

In the case of a real interval of possible outcomes, symmetry with respect to interchanging sub-intervals of equal length corresponds to a continuous uniform distribution.

In other cases, such as "taking a random integer" or "taking a random real number", there are no probability distributions at all symmetric with respect to relabellings or to exchange of equally long subintervals. Other reasonable symmetries do not single out one particular distribution, or in other words, there is not a unique probability distribution providing maximum symmetry.

There is one type of isometry in one dimension that may leave the probability distribution unchanged, that is reflection in a point, for example zero.

A possible symmetry for randomness with positive outcomes is that the former applies for the logarithm, i.e., the outcome and its reciprocal have the same distribution. However this symmetry does not single out any particular distribution uniquely.

For a "random point" in a plane or in space, one can choose an origin, and consider a probability distribution with circular or spherical symmetry, respectively.

## Skew-symmetry

A function of two variables is skew-symmetric if f(y, x) = −f(x, y). The property implies f(x, x) = 0 (except in fields of characteristic two). A skew-symmetric matrix, seen as a function of the row- and column number, is an example.

The property is also called antisymmetry and, in the case of operator notation, anticommutativity.

In the definition of an antisymmetric relation, "minus" is replaced by "not", and the condition is necessarily relaxed, to be required only in the case xy. The corresponding 2D set has a special kind of geometric "symmetry".

More generally, a figure may be such that a particular involution (reflection in a point or line, or e.g. a circle reflection) interchanges e.g. black and white. For example, this applies for the taijitu (symbol of yin and yang) with respect to point inversion.

## Symmetry in probability theory

In probability theory, from a symmetry in stochastic events, a corresponding symmetry of the probability distribution may be derived. For example, due to the approximate symmetry of a die each outcome of tossing one, in the sample space {1, 2, 3, 4, 5, 6}, has approximately the same probability.

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The definition of symmetry needs to be memorized.  If a shape has a line of symmetry, it means that each side of the line is a "mirror image" of the other.  Think about a butterfly.  If you draw an imaginary line down the center, and each side will be a mirror image of the other.  We can say that the butterfly is symmetric or symmetrical, and that it has a vertical line of symmetry, since we need to draw an "up-down" line to show that the left half is the mirror image of the right half.

Some shapes have both a vertical as well as a horizontal line of symmetry.  Think about a rectangle.  If you draw a horizontal (left to right) line across the middle of a rectangle, the top half will be the mirror image of the bottom half.  We could also draw a vertical (up-down) line, and we'd see that the left half of the rectangle would be the same as the right half.  We can say that the rectangle has two lines of symmetry.

One way to help determine if a shape has a line of symmetry is to draw what you think is a line of symmetry, and then fold the shape along that line.  The two sides should be exact mirror images of each other.  If you do this with a symmetrical shape drawn on thin paper, you might be able to see that when one side is folded directly on top of the other, they will line up perfectly.

Sometimes a test question will show you some capital letters of the alphabet, and ask you to determine which ones have one line of symmetry, and which ones have two.  Certainly many letters have none.  Take a look at these capital letters, and try to see where the lines of symmetry are.  For practice, try it yourself on paper.

• Zero Lines of Symmetry:  F, G, J, L, N, P, Q, R, S, Z
• One Line of Symmetry:  A, B, C, D, E, K, M, T, U, V, W, Y
• Two Lines of Symmetry:  H, I, O, X

Some objects have what is called point symmetry.  What this means is that if you rotate the object 180° (a half-circle) around a point, it will look the same.  Stated another way, it looks the same upside-down.  A good example is the letter S and the letter Z.  See if you can think of some other examples.

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The word symmetry orginates from the Latin word symmetria which means measured together.Symmetry means balance or correspondence in size ,shape or arrangement of parts ,things or figures that are in balanced propotion are called symmetrical
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see your maths book or refer the internet
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Symmetry is quite a common word ,used in our daily life. When we see figures with evenly balance proportions , we say ", they are symmetrical.
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Symmetry is quite a common term used in day to day life.when we see certian figures with evenly balancef propotion we say it symmetry
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Equatorial
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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.[a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflectionrotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.

Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objectstheoretic modelslanguagemusic and even knowledge itself.[b]

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 architectureart and music.

The opposite of symmetry is asymmetry.

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.[a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflectionrotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.

In geometry

A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:

• An object has reflectional symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other.
• An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape.
• An object has translational symmetry if it can be translated without changing its overall shape.
• An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.
• An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractalsalso exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions.
• Other symmetries include glide reflection symmetry and rotoreflection symmetry.
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symmetry means a figuere , when folded overlaps itself

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do you want to know what is symetry or what i request you to please post the questions clearly and if you are asking some thing please use a question mark
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Hello dear friend Symmetry is a familiar term used in our day to day life when we see certain figures with evenly balanced proportions ,we say,. They are symmetrical
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Symmetry is a quite a common term used in day to day life. When we see certain figures with evenly balance propotion we say it symmetry
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the state of having two halves that match each other exactly in size, shape, etc.
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Symmetry is a simple line that cuts a given figure into two equal half . It occurs when we fold the figure from middle.
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