Symmetry in mathematics
Encyclopedia
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 transformation
s.
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 function
s, the value of the output is invariant under permutation
s 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.
Note that symmetry is not the exact opposite of antisymmetry
.
More on Symmetric relation
.
s, the value of the output is invariant under permutation
s 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
Some symmetries clear from the problem can be verified in the formula; the distance is invariant under:
.
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
.
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 action
s). 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.
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 equation
s, 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.
.
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.
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.
, 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.
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
occurs not only in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, but also in other branches of mathematics. It is actually the same as invariance
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
: the property that something does not change under a set of transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...
s.
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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
.
In the case of symmetric function
Symmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
s, the value of the output is invariant under permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of variables. These permutations form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
. In the case of isometric
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
transformations in Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, one uses the term symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
. 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
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
.
More on Symmetric relation
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
.
Symmetric functions
In the case of symmetric functionSymmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
s, the value of the output is invariant under permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s 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 distanceGreat-circle distanceThe great-circle distance or orthodromic distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere . Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a...
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 derivativesSymmetry of second derivatives
In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...
.
A relation is symmetric
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
if and only if the corresponding boolean-valued function
Boolean-valued function
A boolean-valued function, in some usages is a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain....
is a symmetric function.
A binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
is commutative if the operator, as function of two variables, is a symmetric function. Symmetric operators on sets include the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
, intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and symmetric difference
Symmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
.
The whole subject of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
deals with well-hidden symmetries of fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
.
A high-level concept related to symmetry is mathematical duality
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
.
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 notationPoint groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...
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 action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s). 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 equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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
Reduction of order
Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y_1 is known and a second linearly independent solution y_2 is desired.-An Example:...
.
For ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, 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 relationEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
.
Randomness
The idea of randomness, without clauses, suggests a probability distributionProbability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
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
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
.
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
Symmetry groups in one dimension
A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension .A pattern in 1D can be represented as a function f for, say, the color at position x....
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
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
two). A skew-symmetric matrix
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
, seen as a function of the row- and column number, is an example.
The property is also called antisymmetry
Antisymmetric
The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...
and, in the case of operator notation, anticommutativity
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
.
In the definition of an antisymmetric relation
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
, "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
Taijitu
Taijitu is a term which refers to a Chinese symbol for the concept of yin and yang...
(symbol of yin and yang
Yin and yang
In Asian philosophy, the concept of yin yang , which is often referred to in the West as "yin and yang", is used to describe how polar opposites or seemingly contrary forces are interconnected and interdependent in the natural world, and how they give rise to each other in turn. Opposites thus only...
) with respect to point inversion.
Symmetry in probability theory
In probability theoryProbability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, 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
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
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)