Symmetry in mathematics

For other uses, see Symmetry (disambiguation) and Bilateral (disambiguation).

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 relation

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 $d(\theta_1, \varphi_1, \theta_2, \varphi_2) = r \cos^{-1}\left(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2\cos(\varphi_1 - \varphi_2)\right).$

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

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.

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^[1].

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^[1]. 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

taijitu

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 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 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.

See also

• Use of symmetry in integration
• Invariance (mathematics)

References

1. ^ ^{a b} Olver, Peter J. (1986). Applications of Lie Groups to Differential Equations. New York: Springer Verlag. ISBN 978-0-387-95000-6. http://books.google.co.uk/books?id=sI2bAxgLMXYC&lpg=PP1&ots=cZKfSvIB1X&dq=Applications%20of%20Lie%20Groups%20to%20Differential%20Equations&pg=PA24#v=onepage&q=&f=false. 

Bibliography

• Hermann Weyl, Symmetry. Reprint of the 1952 original. Princeton Science Library. Princeton University Press, Princeton, NJ, 1989. viii+168 pp. ISBN 0-691-02374-3
• Mark Ronan, Symmetry and the Monster, Oxford University Press, 2006. ISBN 978-0-19-280723-6 (Concise introduction for lay reader)
• Marcus du Sautoy, Finding Moonshine: a Mathematician's Journey through Symmetry, Fourth Estate, 2009