What is mathematical symmetry?
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that something does not change under a set of transformations.
Discrete symmetry
In mathematics, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called "reflections" or "interchanges". 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.
Symmetry of diatomic molecules
The physical laws governing a system is generally written as a relation (equations, differential equations, integral equations etc.). An operation on the ingredients of this relation, which keeps the form of the relations invariant is called a symmetry transformation or a symmetry of the system.
Symmetry is a fundamentally important concept in quantum mechanics. It can predict conserved quantities and provide quantum numbers. It can predict degeneracies of eigenstates and gives insights about the matrix elements of the Hamiltonian without calculating them. Rather than looking into individual symmetries, it is sometimes more convenient to look into the general relations between the symmetries. It turns out that Group theory is the most efficient way of doing this.
Uniform tiling symmetry mutations
In geometry, a symmetry mutation is a mapping of fundamental domains between two symmetry groups. They are compactly expressed in orbifold notation. These mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases.
Symmetry
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article "More is Different" that "it is only slightly overstating the case to say that physics is the study of symmetry." 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.
Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions. Many university level textbooks on physical chemistry, quantum chemistry, and inorganic chemistry devote a chapter to symmetry.
Gauge symmetry (mathematics)
In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.
Matrix (mathematics)
A square matrix A that is equal to its transpose, that is, A = A, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −A, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A.
Mathematics and art
Symmetries are prominent in textile arts including quilting, knitting, cross-stitch, crochet, embroidery and weaving, where they may be purely decorative or may be marks of status. Rotational symmetry is found in circular structures such as domes; these are sometimes elaborately decorated with symmetric patterns inside and out, as at the 1619 Sheikh Lotfollah Mosque in Isfahan. Items of embroidery and lace work such as tablecloths and table mats, made using bobbins or by tatting, can have a wide variety of reflectional and rotational symmetries which are being explored mathematically.
Symmetry of diatomic molecules
where
Homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
Symmetry group
The "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a group action on objects in it.
Symmetry (physics)
formula_21
Visual space
Geometry is the discipline devoted to the study of space and the rules relating the elements to each other. For example, in Euclidean space there is the Pythagorean theorem for distances. In a two-dimensional space of constant curvature, like the surface of a sphere, the rule is somewhat more complex but applies everywhere. On the two-dimensional surface of a football, the rule is more complex still and has different values depending on location. In well-behaved spaces such rules used for measurement and called "Metrics," are classically handled by the mathematics invented by Riemann. Object space belongs to that class.
Symmetry in mathematics
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.
Elementary mathematics
Solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. Stereometry deals with the measurements of volumes of various solid figures (three-dimensional figures) including pyramids, cylinders, cones, truncated cones, spheres, and prisms.
Symmetry (geometry)
For each line or plane of reflection, the symmetry group is isomorphic with C (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically isomorphic to C. The fundamental domain is a half-plane or half-space.
International Society for the Interdisciplinary Study of Symmetry
The International Symmetry Society ("International Society for the Interdisciplinary Study of Symmetry", abbreviated name SIS) is a non-profit organization registered in Hungary (Budapest, Vármegye u. 7. II. 3., H-1052).
Its main objectives are:
Symmetry group
In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme.
Symmetry (physics)
A family of particular transformations may be "continuous" (such as rotation of a circle) or "discrete" (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).
Symmetry in mathematics
In abstract algebra, an automorphism is an isomorphism from a mathematical object to itself. 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. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.In quantum mechanics, bosons have representatives that are symmetric under permutation operators, and fermions have antisymmetric representatives.
Time translation symmetry
Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation. If a symmetry is preserved under a transformation it is said to be "invariant". Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.
Symmetry operation
In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state.
Two basic facts follow from this definition, which emphasize its usefulness.
Wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.
Rotational symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks the same.
Symmetry in mathematics
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.
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional continuous symmetry, e.g. reflection of a 2 dimensional object in 3 dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.
Circular symmetry
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Symmetry in mathematics
or
Translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation.
Symmetry in mathematics
is necessarily antisymmetric. To prove it, consider the matrix element:
Group theory
Given a structured object "X" of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
Symmetry in mathematics
The following 3×3 matrix is symmetric:
Mirror symmetry (string theory)
In physics, mirror symmetry is justified on physical grounds. However, mathematicians generally require rigorous proofs that do not require an appeal to physical intuition. From a mathematical point of view, the version of mirror symmetry described above is still only a conjecture, but there is another version of mirror symmetry in the context of topological string theory, a simplified version of string theory introduced by Edward Witten, which has been rigorously proven by mathematicians. In the context of topological string theory, mirror symmetry states that two theories called the A-model and B-model are equivalent in the sense that there is a duality relating them. Today mirror symmetry is an active area of research in mathematics, and mathematicians are working to develop a more complete mathematical understanding of mirror symmetry based on physicists' intuition.
Symmetry in biology
Biradial symmetry is a combination of radial and bilateral symmetry, as in the ctenophores. Here, the body components are arranged with similar parts on either side of a central axis, and each of the four sides of the body is identical to the opposite side but different from the adjacent side. This may represent a stage in the evolution of bilateral symmetry "from a presumably radially symmetrical ancestor."
Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in more general contexts as expanded below.
Symmetry in mathematics
In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments:
for every permutation σ of the symbols {1,2...,"r"}.
Alternatively, an "r" order symmetric tensor represented in coordinates as a quantity with "r" indices satisfies
Euclidean space
Symmetries of a Euclidean space are transformations which preserve the Euclidean metric (called "isometries"). Although aforementioned translations are most obvious of them, they have the same structure for any affine space and do not show a distinctive character of Euclidean geometry. Another family of symmetries leave one point fixed, which may be seen as the origin without loss of generality. All transformations, which preserves the origin and the Euclidean metric, are linear maps. Such transformations must, for any and , satisfy:
Introduction to systolic geometry
Thus, in the plane central symmetry is the rotation by 180 degrees. For example, an ellipse is centrally symmetric, as is any ellipsoid in 3-space.
Symmetry (geometry)
A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines Euclidean geometry, whereas the group of Möbius transformations defines projective geometry.
Symmetry in mathematics
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.
Symmetry (geometry)
A geometric object has symmetry if there is an "operation" or "transformation" (such as an isometry or affine map) that maps the figure/object onto itself; i.e., it is said that the object has an invariance under the transform. For instance, a circle rotated about its center will have the same shape and size as the original circle—all points before and after the transform would be indistinguishable. A circle is said to be "symmetric under rotation" or to have "rotational symmetry". If the isometry is the reflection of a plane figure, the figure is said to have reflectional symmetry or line symmetry; moreover, it is possible for a figure/object to have more than one line of symmetry.
Symmetry in quantum mechanics
where r′ are the rotated coordinates about an axis defined by a unit vector formula_27 through an angular increment Δ"θ", given by:
Symmetry in mathematics
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Rotational symmetry
In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal:
Dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dih ( "n" ≥ 2 ).
Symmetry
Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is "symmetric" with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group.
Symmetry set
The symmetry set is then the set of formula_21 such that there exist formula_22 with formula_23, and 
together with the limiting points of this set.
International Society for the Interdisciplinary Study of Symmetry
The topic was introduced for the first time by Russian and Polish scholars. Then in 1952, Hermann Weyl published his fascinating book "Symmetry", which was later translated into 10 languages. Since then, it became an attractive subject of research in various fields. A variety of manifestations of the principle of symmetry in sculpture, painting, architecture, ornament, and design, in organic and inorganic nature has been revealed; the philosophical and mathematical significance of this principle has been studied.
During the 1980's the discussions concerning the nature of the world, whether it was essentially probabilistic or naturally geometric, revived the interest of the researchers to the topic. The intellectual atmosphere of this period facilitated the idea of establishment of a new institution devoted to the study of all forms of complexity and patterns of symmetry and orderly structures pervading science, nature and society, that ultimately led to the establishment of the International Society for the Interdisciplinary Study of Symmetry.
Symmetry (physics)
A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called "reflections" or "interchanges".The Standard model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe. CP violation is a fruitful area of current research in particle physics.
Continuous symmetry
The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes continuous symmetry is modelled by a "group action" of a topological group that preserves some structure. Particularly, let formula_1 be a function, and "G" is a group that acts on "X" then a subgroup formula_2 is a symmetry of "f" if formula_3 for all formula_4."
Molecular symmetry
This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarizes all symmetry operations that all molecules in that category have. The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.
Symmetric cone
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by . The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.
Mirror symmetry (string theory)
Many of the important mathematical applications of mirror symmetry belong to the branch of mathematics called enumerative geometry. In enumerative geometry, one is interested in counting the number of solutions to geometric questions, typically using the techniques of algebraic geometry. One of the earliest problems of enumerative geometry was posed around the year 200 BCE by the ancient Greek mathematician Apollonius, who asked how many circles in the plane are tangent to three given circles. In general, the solution to the problem of Apollonius is that there are eight such circles.
Enumerative problems in mathematics often concern a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic (see the illustration) is defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.
Symmetry in mathematics
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.
Symmetry (physics)
formula_17
Special relativity
and this is the physical symmetry underlying special relativity.
Facial symmetry
Facial symmetry is one specific measure of bodily asymmetry. Along with traits such as averageness and youthfulness it influences judgments of aesthetic traits of physical attractiveness and beauty. For instance, in mate selection, people have been shown to have a preference of symmetry. This is because it is seen an indicator of health and genetic fitness, but also as holding adaptation qualities; reflecting the ability to withstand the changes in their environments.
Symmetry (geometry)
Scale symmetry means that if an object is expanded or reduced in size, the new object has the same properties as the original. This is "not" true of most physical systems, as witness the difference in the shape of the legs of an elephant and a mouse (so-called allometric scaling). Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight.
Plane symmetry
A plane symmetry is a symmetry of a pattern in the Euclidean plane: that is, a transformation of the plane that carries any directioned lines to lines and preserves many different distances. If one has a pattern in the plane, the set of plane symmetries that preserve the pattern forms a group. The groups that arise in this way are plane symmetry groups and are of considerable mathematical interest. A symmetry plane is a three-dimensional object's symmetry axe.
Asymmetry
Symmetry is one of the most powerful tools in particle physics, because it has become evident that practically all laws of nature originate in symmetries. Violations of symmetry therefore present theoretical and experimental puzzles that lead to a deeper understanding of nature. Asymmetries in experimental measurements also provide powerful handles that are often relatively free from background or systematic uncertainties.
Skewness
A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000. It is called distance skewness and denoted by dSkew. If "X" is a random variable taking values in the "d"-dimensional Euclidean space, "X" has finite expectation, "X" is an independent identically distributed copy of "X", and formula_31 denotes the norm in the Euclidean space, then a simple "measure of asymmetry" with respect to location parameter θ is
and dSkew("X") := 0 for "X" = θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if "X" is diagonally symmetric with respect to θ ("X" and θ −"X" have the same probability distribution) and equals 1 if and only if X is a constant "c" (formula_33) with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness:
Symmetry (physics)
The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.
Gauge symmetry (mathematics)
A gauge symmetry of a Lagrangian formula_1 is defined as a differential operator on some vector bundle formula_2 taking its values in the linear space of (variational or exact) symmetries of formula_1. Therefore, a gauge symmetry of formula_1
depends on sections of formula_2 and their partial derivatives. For instance, this is the case of gauge symmetries in classical field theory. Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.
Symmetry
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:A dyadic relation "R" is symmetric if and only if, whenever it's true that "Rab", it's true that "Rba". Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul.
Event symmetry
General relativity by contrast, is a model theory based on a geometric symmetry principle from which its dynamics can be elegantly derived. The symmetry is called general covariance or diffeomorphism invariance. It says that the dynamical equations of the gravitational field and any matter must be unchanged in form under any smooth transformation of spacetime coordinates. To understand what that means you have to think of a region of spacetime as a set of "events", each one labelled by unique values of four coordinate values x,y,z, and t. The first three tell us "where" in space the event happened, while the fourth is time and tells us "when" it happened. But the choice of coordinates that are used is arbitrary, so the laws of physics should not depend on what the choice is. It follows that if any smooth mathematical function is used to map one coordinate system to any other, the equations of dynamics must transform in such a way that they look the same as they did before. This symmetry principle is a strong constraint on the possible range of equations and can be used to derive the laws of gravity almost uniquely.
Symmetric matrix
Since this definition is independent of the choice of basis, symmetry is a property that depends only on the linear operator A and a choice of inner product. This characterization of symmetry is useful, for example, in differential geometry, for each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian manifold. Another area where this formulation is used is in Hilbert spaces.
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. 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. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.
Symmetry (physics)
The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by continuous or smooth functions. An important subclass of continuous symmetries in physics are spacetime symmetries.
Symmetry (physics)
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
Symmetry in biology
Symmetry in biology is the balanced distribution of duplicate body parts or shapes within the body of an organism. In nature and biology, symmetry is always approximate. For example, plant leaves – while considered symmetrical – rarely match up exactly when folded in half. Symmetry creates a class of patterns in nature, where the near-repetition of the pattern element is by reflection or rotation.
Symmetry (geometry)
Rotational symmetry is symmetry with respect to some or all rotations in -dimensional Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean group E().
Symmetry (physics)
The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, the isometry of space gives rise to conservation of (linear) momentum, and isometry of time gives rise to conservation of energy.
Symmetry in quantum mechanics
where is a finite-dimensional representation, in other words a dimensional square matrix, and is thought of as a column vector containing components with the allowed values of :
Local symmetry
In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory. These kinds of symmetries, also known as internal symmetries, are distinguished from spacetime symmetries.
Symmetry group
A discrete symmetry group is a symmetry group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. The number of elements in the group may be either finite or infinite.
Symmetry (disambiguation)
Other uses:
Symmetry in mathematics
An isometry is a distance-preserving map between metric spaces. Given a metric space, or a set and scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to another metric space such that the distance between the elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional space, two geometric figures are congruent if they are related by an isometry: related by either a rigid motion, or a composition of a rigid motion and a reflection. Up to a relation by a rigid motion, they are equal if related by a direct isometry.
Symmetry (geometry)
The most common group of transforms applied to objects are termed the Euclidean group of "isometries," which are distance-preserving transformations in space commonly referred to as two-dimensional or three-dimensional (i.e., in plane geometry or solid geometry Euclidean spaces). These isometries consist of reflections, rotations, translations, and combinations of these basic operations. Under an isometric transformation, a geometric object is said to be symmetric if, after transformation, the object is indistinguishable from the object before the transformation. A geometric object is typically symmetric only under a subset or "subgroup" of all isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups and by the types of object invariance that are possible in geometry.
Symmetry (disambiguation)
In arts and entertainment:
Symmetry in quantum mechanics
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.
Circular symmetry
In 3-dimensions, a surface or solid of revolution has circular symmetry around an axis, also called cylindrical symmetry. For example: a right circular cone. Circular symmetry in 3 dimensions has all pyramidal symmetry, C as subgroups.
Symmetric set
In mathematics, a nonempty subset "S" of a group "G" is said to be symmetric if
where formula_2. In other words, "S" is symmetric if formula_3 whenever formula_4.
Symmetry in mathematics
A Line symmetry of a system of differential equations is a continuous symmetry of the system of differential equations. Knowledge of a Line symmetry can be used to simplify an ordinary differential equation through reduction of order.
Symmetry in mathematics
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric.
Symmetry in quantum mechanics
where "σ" is a Pauli matrix, and the group parameters are the angles turned through about an axis.
Symmetry in mathematics
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.
Aesthetics
Mathematical considerations, such as symmetry and complexity, are used for analysis in theoretical aesthetics. This is different from the aesthetic considerations of applied aesthetics used in the study of mathematical beauty. Aesthetic considerations such as symmetry and simplicity are used in areas of philosophy, such as ethics and theoretical physics and cosmology to define truth, outside of empirical considerations. Beauty and Truth have been argued to be nearly synonymous, as reflected in the statement "Beauty is truth, truth beauty" in the poem Ode on a Grecian Urn by John Keats, or by the Hindu motto "Satyam Shivam Sundaram" (Satya (Truth) is Shiva (God), and Shiva is Sundaram (Beautiful)). The fact that judgments of beauty and judgments of truth both are influenced by processing fluency, which is the ease with which information can be processed, has been presented as an explanation for why beauty is sometimes equated with truth. Indeed, recent research found that people use beauty as an indication for truth in mathematical pattern tasks. However, scientists including the mathematician David Orrell and physicist Marcelo Gleiser have argued that the emphasis on aesthetic criteria such as symmetry is equally capable of leading scientists astray.
Ordinary differential equation
Symmetry methods have been recognized to study differential equations, arising in mathematics, physics, engineering, and many other disciplines.
Symmetry
Symmetry is important to chemistry because it undergirds essentially all "specific" interactions between molecules in nature (i.e., via the interaction of natural and human-made chiral molecules with inherently chiral biological systems). 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. A rigorous understanding of symmetry explains fundamental observations in quantum chemistry, and in the applied areas of spectroscopy and crystallography. The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory.
Similarity (geometry)
Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set of numbers of the form where ranges over all integers. When this set is plotted on a logarithmic scale it has one-dimensional translational symmetry: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.
Symmetry
In biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialized with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.
Rotational symmetry
Formally the rotational symmetry is symmetry with respect to some or all rotations in "m"-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of "E"("m") (see Euclidean group).
Symmetry (physics)
Continuous "spacetime symmetries" are symmetries involving transformations of space and time. These may be further classified as "spatial symmetries", involving only the spatial geometry associated with a physical system; "temporal symmetries", involving only changes in time; or "spatio-temporal symmetries", involving changes in both space and time.Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.
Symmetry in mathematics
A symmetric polynomial is a polynomial "P"("X", "X", …, "X") in "n" variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, "P" is a "symmetric polynomial", if for any permutation σ of the subscripts 1, 2, ..., "n" one has "P"("X", "X", …, "X") = "P"("X", "X", …, "X").
Symmetry in mathematics
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.
Symmetry (disambiguation)
In science and mathematics:
Event symmetry
It is not immediately obvious how event symmetry could work. It seems to say that taking one part of space time and swapping it with another part a long distance away is a valid physical operation, and that the laws of physics must be written to support this. Clearly this symmetry can only be correct if it is hidden or broken. To get this in perspective consider what the symmetry of general relativity seems to say. A smooth coordinate transformation or diffeomorphism can stretch and twist spacetime in any way so long as it is not torn. The laws of general relativity are unchanged in form under such a transformation. Yet this does not mean that objects can be stretched or bent without being opposed by a physical force. Likewise, event symmetry does not mean that objects can be torn apart in the way the permutations of spacetime would make us believe. In the case of general relativity the gravitational force acts as a background field that controls the measurement properties of spacetime. In ordinary circumstances the geometry of space is flat and Euclidean and the diffeomorphism invariance of general relativity is hidden thanks to this background field. Only in the extreme proximity of a violent collision of black holes would the flexibility of spacetime become apparent. In a similar way, event symmetry could be hidden by a background field that determines not just the geometry of spacetime, but also its topology.
Psychology of art
Symmetry and beauty have a strong biological link that influences aesthetic preferences. It has been shown that humans tend to prefer art that contains symmetry, deeming it more beautiful. Furthermore, symmetry directly correlates to the understanding of a face or artwork as beautiful. The greater the symmetry within the work or the face, generally the more beautiful it appears to be. Research on aesthetic preference for geometric forms and the fluent processing of symmetry sheds light on the role that symmetry plays in the overall aesthetic judgment and experience.
Group (mathematics)
"Symmetry groups" are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object "X" if every group element performs some operation on "X" compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.
Symmetry in mathematics
The integral of an even function from −"A" to +"A" is twice the integral from 0 to +"A" (where "A" is finite, and the function has no vertical asymptotes between −"A" and "A". This also holds true when "A" is infinite, but only if the integral converges).In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix "A" is symmetric if