Symmetry in Mathematics

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Symmetry in mathematics 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.

A function of n variables may be invariant under certain permutations of the variables. These permutations form a group, a symmetry group.

Examples:

• 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 the case of a symmetric function, all permutations give the same value. A symmetric matrix, seen as a 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 iff the corresponding boolean-valued function is a symmetric function.

An 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.

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, for the equation (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).

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 C_3v. If only two unknowns could be interchanged this would be C_s.

More generally one can also consider other objects than functions and equations, and other operations than permutations of variables that leave the object unchanged. Again these operations form a group; for an algebraic object, one uses the term automorphism group instead of symmetry group. The whole subject of Galois theory deals with well-hidden symmetries of fields. In fact, prior to the 20^th century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20^th century that the current abstract definition of a group without any reference to group actions was used instead.

Symmetry can itself be invariant under a group of transformations. See the diamond theorem.

A concept related to symmetry is duality (mathematics).

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 them implies a discrete uniform distribution.

In the case of a real interval of possible outcomes, maximum symmetry with respect to them corresponds to a continuous uniform distribution.

In other cases, such as "taking a random integer" or "taking a random real number", only little symmetry is possible, there is not a particular probability distribution providing maximum symmetry, so that probability distribution should be specified.

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.

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

yin and yang symbol

A function of two variables is skew-symmetric if f(y,x) = - f(x,y). The property implies f(x,x) = 0. 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 x ≠ y. 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 yin and yang symbol with respect to point inversion.

Symmetry in probability theory

In probability theory, from symmetry of a sample space corresponding symmetry of the probability distribution may be derived.