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 invaria
Group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homo
The Symmetries of Things
The Symmetries of Things is a book on mathematical symmetry and the symmetries of geometric objects, aimed at audiences of multiple levels. It was written over the course of many years by John Horton
Translational symmetry
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by a: Ta(p) = p + a. In physics and mathematics, continuous tra
Cymatics
Cymatics (from Ancient Greek: κῦμα, romanized: kyma, lit. 'wave') is a subset of modal vibrational phenomena. The term was coined by Hans Jenny (1904-1972), a Swiss follower of the philosophical schoo
Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In p
Arch form
In music, arch form is a sectional structure for a piece of music based on repetition, in reverse order, of all or most musical sections such that the overall form is symmetric, most often around a ce
Centrosymmetry
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistingu
Modular invariance
In theoretical physics, modular invariance is the invariance under the group such as SL(2,Z) of large diffeomorphisms of the torus. The name comes from the classical name modular group of this group,
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that parti
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 s
Time translation symmetry
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the
Non-Euclidean crystallographic group
In mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in
Explicit symmetry breaking
In theoretical physics, explicit symmetry breaking is the breaking of a symmetry of a theory by terms in its defining equations of motion (most typically, to the Lagrangian or the Hamiltonian) that do
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived fr
Time reversibility
A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed. A deterministic process is time-reversible if the
List of finite spherical symmetry groups
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral,
Schoenflies notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point grou
Higgs sector
In particle physics, the Higgs sector is the collection of quantum fields and/or particles that are responsible for the Higgs mechanism, i.e. for the spontaneous symmetry breaking of the Higgs field.
3D mirror symmetry
In theoretical physics, 3D mirror symmetry is a version of mirror symmetry in 3-dimensional gauge theories with N=4 supersymmetry, or 8 supercharges. It was first proposed by and Nathan Seiberg, in th
Symmetry element
In chemistry and crystallography, a symmetry element is a point, line, or plane about which symmetry operations can take place. In particular, a symmetry element can be a mirror plane, an axis of rota
Transformation geometry
In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and
Droste effect
The Droste effect (Dutch pronunciation: [ˈdrɔstə]), known in art as an example of mise en abyme, is the effect of a picture recursively appearing within itself, in a place where a similar picture woul
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation
Tendril perversion
Tendril perversion is a geometric phenomenon sometimes observed in helical structures in which the direction of the helix transitions between left-handed and right-handed. Such a reversal of chirality
Symmetry
Symmetry (from Ancient Greek: συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(N), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behav
Symmetric spectrum
In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group on such that the composition of structure maps is equivariant wi
Screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displa
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of
Through and through
Through and through describes a situation where an object, real or imaginary, passes completely through another object, also real or imaginary. The phrase has several common uses:
Family symmetries
In particle physics, the family symmetries or horizontal symmetries are various discrete, global, or local symmetries between quark-lepton families or generations. In contrast to the intrafamily or ve
Space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid tra
Symmetry number
The symmetry number or symmetry order of an object is the number of different but indistinguishable (or equivalent) arrangements (or views) of the object, that is, it is the order of its symmetry grou
The Ambidextrous Universe
The Ambidextrous Universe is a popular science book by Martin Gardner, covering aspects of symmetry and asymmetry in human culture, science and the wider universe. It culminates in a discussion of whe
Jucys–Murphy element
In mathematics, the Jucys–Murphy elements in the group algebra of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: The
Rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, 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 t
Affine symmetric group
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensiona
Conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformatio
List of space groups
There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with space
Weyl transformation
In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: which produces another metric in the same conformal class. A theory or an expressi
Molecular symmetry
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemis
Lorentz covariance
In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity i
Polar point group
In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. The unmoved points will constitute a line, a plane, or all of spac
Zimmer's conjecture
Zimmer's conjecture is a statement in mathematics "which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries." It was named after the mathematician Robert
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacet
Higgs field (classical)
Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breakin
Jay Hambidge
Jay Hambidge (1867–1924) was a Canadian-born American artist who formulated the theory of "dynamic symmetry", a system defining compositional rules, which was adopted by several notable American and C
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 a mathematical object remains unchanged under a set of operations o
Symmetry (geometry)
In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an i
Symmetry operation
In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that
Triptych
A triptych (/ˈtrɪptɪk/ TRIP-tik; from the Greek adjective τρίπτυχον "triptukhon" ("three-fold"), from tri, i.e., "three" and ptysso, i.e., "to fold" or ptyx, i.e., "fold") is a work of art (usually a
Lie group
In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract c
Fock–Lorentz symmetry
Lorentz invariance follows from two independent postulates: the principle of relativity and the principle of constancy of the speed of light. Dropping the latter while keeping the former leads to a ne
Gul (design)
A gul (also written gol, göl and gül) is a medallion-like design element typical of traditional hand-woven carpets from Central and West Asia. In Turkmen weavings they are often repeated to form the p
Symmetry of diatomic molecules
Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in
Noether's second theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of
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 o
Scale invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus
Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a transl
Inequivalent symmetry
Two symmetrical patterns are considered to be if they have exactly the same types of symmetry. As recently as 1891, it was finally proved that there are only 17 inequivalent symmetry patterns in the p
CPT symmetry
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T)
Symmetry-protected topological order
Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-inva
Geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and
Stueckelberg action
In field theory, the Stueckelberg action (named after Ernst Stueckelberg) describes a massive spin-1 field as an R (the real numbers are the Lie algebra of U(1)) Yang–Mills theory coupled to a real sc
Conformal symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatia
Spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describ
Confronted animals
Confronted animals, or confronted-animal as an adjective, where two animals face each other in a symmetrical pose, is an ancient bilateral motif in art and artifacts studied in archaeology and art his
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function of n va
Symmetry in biology
Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take
Inversion transformation
In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal one-to-one transformations on coordinate space-time. They are less studi
Elitzur's theorem
In quantum field theory and statistical field theory, Elitzur's theorem states that in gauge theories, the only operators that can have non-vanishing expectation values are ones that are invariant und
Equivariant map
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and cod
Bondi–Metzner–Sachs group
In gravitational theory, the Bondi–Metzner–Sachs (BMS) group, or the Bondi–van der Burg–Metzner–Sachs group, is an asymptotic symmetry group of asymptotically flat, Lorentzian spacetimes at null (i.e.
Facial symmetry
Facial symmetry is one specific measure of bodily symmetry. Along with traits such as averageness and youthfulness it influences judgments of aesthetic traits of physical attractiveness and beauty. Fo
Symmetry breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifu
Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation express
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 funct
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. A family of pa
C-symmetry
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the char
Directional symmetry (time series)
In statistical analysis of time series and in signal processing, directional symmetry is a statistical measure of a model's performance in predicting the direction of change, positive or negative, of
SYZ conjecture
The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and
Murnaghan–Nakayama rule
In group theory, a branch of mathematics, the Murnaghan–Nakayama rule is a combinatorial method to compute irreducible character values of a symmetric group.There are several generalizations of this r
Discrete symmetry
In mathematics and geometry, 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 mul
P-compact group
In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was
Lie point symmetry
Lie point symmetry is a concept in advanced mathematics. Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differenti
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element
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. Rotational circular symmetry is isomorphic with the
One-dimensional symmetry group
A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function f(x) for, say, the color at position x. The o
Conserved current
In physics a conserved current is a current, , that satisfies the continuity equation . The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equati
Einstein group
Albert Einstein, in searching for the transformation group for his unified field theory, wrote: Every attempt to establish a unified field theory must start, in my opinion, from a group of transformat
Glide plane
In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crys
Misorientation
In materials science, misorientation is the difference in crystallographic orientation between two crystallites in a polycrystalline material. In crystalline materials, the orientation of a crystallit
Fibrifold
In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al., who
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, Since the second law of thermodynamics states that entropy increases as tim
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, whe
Introduction to gauge theory
A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a
Symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformatio
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abs
Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs
Cyclic symmetry in three dimensions
In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does
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 an abstract group is a dihedral group Dihn
Supersymmetry breaking
In particle physics, supersymmetry breaking is the process to obtain a seemingly non-supersymmetric physics from a supersymmetric theory which is a necessary step to reconcile supersymmetry with actua
Axiality (geometry)
In the geometry of the Euclidean plane, axiality is a measure of how much axial symmetry a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the wh
Lie groupoid
In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map a
Hesse's principle of transfer
In geometry, Hesse's principle of transfer (German: Übertragungsprinzip) states that if the points of the projective line P1 are depicted by a rational normal curve in Pn, then the group of the projec
Soft SUSY breaking
In theoretical physics, soft SUSY breaking is type of supersymmetry breaking that does not cause ultraviolet divergences to appear in scalar masses.
Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontri
Crystal system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classifie
Triskelion
A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry.The spiral design can be based on interlocking Archimedean spirals, or represent three bent h