Sphere symmetrical group o. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe
Leonardo da Vinci's Vitruvian Man (ca. Leonardo di ser Piero da Vinci ( April 15 1452 – May 2 1519 was an Italian Polymath, having been a scientist Mathematician, Engineer The Vitruvian Man is a world-renowned Drawing with accompanying notes created by Leonardo da Vinci around the year 1487 as recorded in one of his journals 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.

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. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Although the meanings are distinguishable, in some contexts, both meanings of "symmetry" are related and discussed in parallel. [1][2]

The "precise" notions of symmetry have various measures and operational definitions. For example, symmetry may be observed:

• with respect to the passage of time;
• as a spatial relationship;
• through geometric transformations such as scaling, reflection, and rotation;
• through other kinds of functional transformations[3]; and
• as an aspect of abstract objects, theoretic models, language, music and even knowledge itself. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In Mathematics, a transformation could be any Function from a set X to itself In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed For other uses see Abstract In Philosophy it is commonly considered that every object is either abstract or concrete Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them Music is an Art form in which the medium is Sound organized in Time. Knowledge is defined ( Oxford English Dictionary) variously as (i expertise and skills acquired by a person through experience or education the theoretical or practical understanding [4][5]

The opposite of symmetry is asymmetry. Asymmetry is the absence of or a violation of a Symmetry. In organisms Due to how cells divide in Organisms asymmetry in organisms is

## Symmetry in the field of mathematics

In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa). S T U V Tragedy of the commons -->

Symmetries may also be found in living organisms including humans and other animals (see symmetry in biology below). In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, and glide reflections. In Geometry, a Euclidean plane isometry is an Isometry of the Euclidean plane, or more informally a way of transforming the plane that preserves geometrical In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation

### Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action g : G × XX, where the image of g in G and x in X is written as g·x. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. If, for some g, g·x = y then x and y are said to be symmetrical to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: VV acting on the set of functions x: VW by (gx)(v)=x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v)=x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v)=w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself.

In a modified version for vector fields, we have (gx)(v)=h(g,x(g−1(v))) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these The symmetry group of x consists of all g for which x(v)=h(g,x(g(v))) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero.

For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional For the Mechanical engineering and Architecture usage see Isometric projection. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. (For generalizations, see the next subsection. ) Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a boolean function of position v), or, at the other extreme, e. g. symmetry of right and left hand with all their structure.

For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, the equivalence classes are the orbits of the group action on the space itself. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry. Asymmetry is the absence of or a violation of a Symmetry. In organisms Due to how cells divide in Organisms asymmetry in organisms is

An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can e. g. :

• take the values in a fundamental domain (i. e. , add copies of the object)
• take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap)

If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric.

As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns").

In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. Current density is a measure of the Density of flow of a conserved charge. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Physics, velocity is defined as the rate of change of Position.

A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {. . . ,1,2,5,6,9,10,13,14,. . . } acts transitively on all these points, while {. . . ,1,2,3,5,6,7,9,10,11,13,14,15,. . . } does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class

### Non-isometric symmetry

As mentioned above, G (the symmetry group of the space itself) may differ from the Euclidean group, the group of isometries. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional For the Mechanical engineering and Architecture usage see Isometric projection.

Examples:

• G is the group of similarity transformations, i. e. affine transformations with a matrix A that is a scalar times an orthogonal matrix. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T Thus dilations are added, self-similarity is considered a symmetry
• G is the group of affine transformations with a matrix A with determinant 1 or -1, i. In Mathematics, a dilation is a function &fnof from a Metric space into itself that satisfies the identity d(f(xf(y=rd(xy \ In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i e. the transformation which preserve area; this adds e. g. oblique reflection symmetry. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect
• G is the group of all bijective affine transformations
• In inversive geometry, G includes circle reflections, etc. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions.
• More generally, an involution defines a symmetry with respect to that involution.

### Directional symmetry

Main article: directional symmetry

### Reflection symmetry

Main article: reflection symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. Directional symmetry is roughly defined as "things going the same direction" and is to be distinguished from directional asymmetry, meaning "things going Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect

In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see mirror image). "Mirror Image" is an episode of the Television series The Twilight Zone.

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides A circle has infinitely many axes of symmetry, for the same reason. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

If the letter T is reflected along a vertical axis, it appears the same. Note that this is sometimes called horizontal symmetry, and sometimes vertical symmetry! One can better use an unambiguous formulation, e. g. "T has a vertical symmetry axis. "

The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. In Geometry a kite, or deltoid, is a Quadrilateral with two disjoint pairs of Congruent Adjacent sides in contrast A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed The fundamental domain is a half-plane or half-space.

Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane. The Bilateria (ˌbaɪləˈtɪəriə are all animals having a bilateral symmetry, i A sagittal plane of the human body is an imaginary plane that travels from the top to the bottom of the body dividing it into left and right portions

In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is equivalent with inversion symmetry; in such contexts in modern physics the term P-symmetry is used for both (P stands for parity).

For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:

• with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc). For the Mechanical engineering and Architecture usage see Isometric projection.

### Rotational symmetry

Main article: rotational symmetry

Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Rotations are direct isometries, i. e. , isometries preserving orientation. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which Therefore a symmetry group of rotational symmetry is a subgroup of E+(m) (see Euclidean group). In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m×m orthogonal matrices with determinant 1. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n For m=3 this is the rotation group. This article is about rotations in three-dimensional Euclidean space

In another meaning of the word, the rotation group of an object is the symmetry group within E+(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has See also rotational invariance. In Mathematics, a function defined on an Inner product space is said to have rotational invariance if its value does not change when arbitrary Rotations

### Translational symmetry

See main article translational symmetry. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a

Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(p) = p + a

### Glide reflection symmetry

A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation It implies translational symmetry with twice the translation vector.

The symmetry group is isomorphic with Z.

### Rotoreflection symmetry

In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or As symmetry groups with regard to a roto-reflection we can distinguish:

• the angle has no common divisor with 360°, the symmetry group is not discrete
• 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n=1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying
• Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point 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

### Helical symmetry

A drill bit with helical symmetry.

Helical symmetry is the kind of symmetry seen in such everyday objects as springs, Slinky toys, drill bits, and augers. The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i A spring is a flexible elastic object used to store mechanical Energy. A Slinky is a Coil -shaped Toy invented by mechanical engineer Richard James in Philadelphia Pennsylvania For the ficitonal character see Drill Bit (Transformers. Drill bits are cutting tools used to create cylindrical holes An auger is a device for moving material or liquid (see Archimedes' screw) by means of a rotating Helical flighting It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis). The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an even angular speed while simultaneously moving at another even speed along its axis of rotation (translation). Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the tracing. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the rotation is slow and the translation speedy, the coiling angle will approach 90°.

Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:

• Infinite helical symmetry. If there are no distinguishing features along the length of a helix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a cross section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. A spring is a flexible elastic object used to store mechanical Energy. A Slinky is a Coil -shaped Toy invented by mechanical engineer Richard James in Philadelphia Pennsylvania For the ficitonal character see Drill Bit (Transformers. Drill bits are cutting tools used to create cylindrical holes An auger is a device for moving material or liquid (see Archimedes' screw) by means of a rotating Helical flighting Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
• n-fold helical symmetry. If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called n-fold helical symmetry, where n = 360°/θ, see e. g. double helix. In Geometry a double helix (plural helices) typically consists of two congruent helices with the same axis differing by a translation This concept can be further generalized to include cases where mθ is a multiple of 360°—that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
• Non-repeating helical symmetry. This is the case in which the angle of rotation θ required to observe the symmetry is an irrational number such as $\sqrt 2$ radians that never repeats exactly no matter how many times the helix is rotated. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 Such symmetries are created by using a non-repeating point group in two dimensions. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed DNA is an example of this type of non-repeating helical symmetry. Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known

### Scale symmetry and fractals

Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does not exist for most physical systems, a point that was first discerned by Galileo. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of elephants versus mice, and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight. Elephants ( family: Elephantidae) are large land Mammals of the order Proboscidea. A mouse (plural mice) is a small Animal that belongs to one

A more subtle form of scale symmetry is demonstrated by fractals. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" As conceived by Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks exactly the same no matter what degree of magnification is used to examine it. Magnification is the process of enlarging something only in appearance not in physical size A coast is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The coast is defined as the part of the land adjoining or near the Ocean. The branching of trees, which enables children to use small twigs as stand-ins for full trees in dioramas, is another example. The word diorama can refer either to a nineteenth century mobile theatre device or in modern usage a three-dimensional model usually enclosed in a glass showcase for a museum

This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the Mandelbrot set. In Mathematics, the Mandelbrot set, named after Benoît Intriguingly, fractals have also found a place in CG, or computer-generated movie effects, where their ability to create very complex curves with fractal symmetries results in more realistic virtual worlds. A virtual world is a computer-based simulated environment intended for its users to inhabit and interact via avatars These avatars are usually depicted

### Symmetry combinations

Main article: symmetry combinations

## Symmetry in science

### Symmetry in physics

Main article: Symmetry in physics

Symmetry in physics has been generalized to mean invariance—that is, lack of any visible change—under any kind of transformation. This article discusses various Symmetry combinations. In 2D mirror-image symmetry in combination with n -fold rotational symmetry with the center of rotational Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these 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. Philip Warren Anderson (born December 13, 1923) is an American Physicist. " See Noether's theorem (which, as a gross oversimplification, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; 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. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Mathematics and Theoretical physics, Wigner's classification is a classification of the Nonnegative Energy irreducible unitary representations

### Symmetry in physical objects

#### Classical objects

Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object.

For example, if one rotates a precisely machined aluminum equilateral triangle 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. Properties The area of an equilateral triangle with sides of length a\\! However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as optical or electron microscopes will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as crystals or minor deformities. The optical microscope, often referred to as the "light microscope" is a type of Microscope which uses Visible light and a system of lenses to An electron microscope is a type of Microscope that uses Electrons to illuminate a specimen and create an enlarged image In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating

Such simple thought experiments show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper symmetries of space and time play a major role in classical physics—that is, the physics of large, everyday objects. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these

#### Quantum objects

Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such as electrons, protons, light, and atoms. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny

Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an observer is. Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.

#### Consequences of quantum symmetry

While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.

However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by Richard Feynman in the direct transcripts of his Feynman Lectures on Physics, Volume III, Section 3. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum The Feynman Lectures on Physics by Richard Feynman, Robert Leighton, and Matthew Sands is perhaps Feynman's most accessible technical work 4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture. )

". . . if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails. "

The word "interferes" in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons A wave is a disturbance that propagates through Space and Time, usually with transference of Energy.

In short, when an object becomes so simple that a symmetry assertion of the form F(x) = x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the rules of classical physics and must instead be modeled using the more complex—and often far less intuitive—rules of quantum physics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.

### Symmetry as a unifying principle of geometry

The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry (an aspect of the New Math, but hardly controversial in modern mathematical practice). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, transformation geometry is a name for a Pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. New Math was a brief dramatic change in the way Mathematics was taught in American Grade schools during the 1960s The name is commonly given By now it has been applied in numerous forms, as kind of standard attack on problems.

### Symmetry in mathematics

An example of a mathematical expression exhibiting symmetry is a²c + 3ab + b²c. Symmetry in Mathematics occurs not only in Geometry, but also in other branches of mathematics If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication. In Mathematics, commutativity is the ability to change the order of something without changing the end result

Like in geometry, for the terms there are two possibilities:

• It is itself symmetric
• It has one or more other terms symmetric with it, in accordance with the symmetry kind

### Symmetry in logic

A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. In Mathematics, the term "symmetric function" can mean two different things In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of 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.

Symmetric binary logical connectives are "and" (∧, $\land$, or &), "or" (∨), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or"). Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements Definition The NAND operation is a Logical operation on two Logical values typically the values of two Propositions that produces a value In Boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the inverse of logical or.

### Generalizations of symmetry

If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, especially in Category theory and Homotopy theory Indeed, A. Connes in his book Non-commutative_geometry' writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum. Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the

The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of higher order symmetry' which have been a little explored, as follows.

The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group G naturally form a crossed module $G \to Aut(G)$, and crossed modules give an algebraic model of homotopy 2-types. In Mathematics, and especially in Homotopy theory, a crossed module consists of groups G and H where G acts on H At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is know to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed n-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics. The web site n-category cafe has much discussion of n-groups. More information is on `Higher dimensional group theory'.

Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems

### Symmetry in biology

See symmetry (biology) and facial symmetry. "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Symmetry, especially facial symmetry, is one of a number of Aesthetic traits including Averageness and Youthfulness, associated with Health

### Symmetry in chemistry

Main article: molecular symmetry

Symmetry is important to chemistry because it explains observations in spectroscopy, quantum chemistry and crystallography. Molecular symmetry in Chemistry describes the Symmetry present in Molecules and the classification of molecules according to their symmetry Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The It draws heavily on group theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.

## Symmetry in history, religion, and culture

In any human endeavor for which an impressive visual result is part of the desired objective, symmetries play a profound role. The innate appeal of symmetry can be found in our reactions to happening across highly symmetrical natural objects, such as precisely formed crystals or beautifully spiraled seashells. Our first reaction in finding such an object often is to wonder whether we have found an object created by a fellow human, followed quickly by surprise that the symmetries that caught out attention are derived from nature itself. In both reactions we give away our inclination to view symmetries both as beautiful and, in some fashion, informative of the world around us.

### Symmetry in religious symbols

Symmetry in religious symbols. Row 1. Christian, Jewish, Hindu Row 2. Christianity ( Greek Χριστιανισμός from the word Xριστός ( Christ)is a monotheistic Religion centered on the life and teachings Judaism (from the Greek Ioudaïsmos, derived from the Hebrew יהודה Yehudah, " Judah " in Hebrew יַהֲדוּת Yahedut Hinduism is a religious tradition that originated in the Indian subcontinent. Islamic, Buddhist, Shinto Row 3. For other meanings including people named 'Islam' see Islam (disambiguation. Buddhism is a family of beliefs and practices is the native religion of Japan and was once its State religion. Sikh, Baha'i, Jain

The tendency of people to see purpose in symmetry suggests at least one reason why symmetries are often an integral part of the symbols of world religions. Sikhism ( IPA: or; ਸਿੱਖੀ sikkhī, IPA:) founded on the teachings of Nanak and nine successive gurus in fifteenth century The Bahá'í Faith is a Religion founded by Bahá'u'lláh in nineteenth-century Persia, emphasizing the spiritual unity of all humankind Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. Just a few of many examples include the sixfold rotational symmetry of Judaism's Star of David, the twofold point symmetry of Taoism's Taijitu or Yin-Yang, the bilateral symmetry of Christianity's cross and Sikhism's Khanda, or the fourfold point symmetry of Jain's ancient (and peacefully intended) version of the swastika. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Judaism (from the Greek Ioudaïsmos, derived from the Hebrew יהודה Yehudah, " Judah " in Hebrew יַהֲדוּת Yahedut The Star of David or Shield of David ( Magen David in Hebrew with nikkud or מגן דוד without academically transcribed Māḡēn Dāwīḏ by In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed Taoism (pronounced /ˈdaʊɪzəm/ or /ˈtaʊɪzəm/ also spelled '''Daoism''') refers to a variety of related Philosophical and Religious traditions In Chinese philosophy, the concept of yin and yang ( is used to describe how seemingly opposing forces are bound together intertwined and interdependent in the "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Christianity ( Greek Χριστιανισμός from the word Xριστός ( Christ)is a monotheistic Religion centered on the life and teachings A cross is a geometrical figure consisting of two lines or bars perpendicular to each other dividing one or two of the lines in half Sikhism ( IPA: or; ਸਿੱਖੀ sikkhī, IPA:) founded on the teachings of Nanak and nine successive gurus in fifteenth century The Khanda ( khaṇḍā) is one of most important symbols of Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. The swastika (from Sanskrit: svástika sa स्वस्तिक Hindu IS CORRECT if 'ि' is positioned incorrectly see -->) is With its strong prohibitions against the use of representational images, Islam, and in particular the Sunni branch of Islam, has developed some of the most intricate and visually impressive use of symmetries for decorative uses of any major religion. For other meanings including people named 'Islam' see Islam (disambiguation. Sunni Islam is the largest denomination of Islam. Sunni Islam is also referred to as Ahl as-Sunnah wa’l-Jamā‘h (Arabic

The ancient Taijitu image of Taoism is a particularly fascinating use of symmetry around a central point, combined with black-and-white inversion of color at opposite distances from that central point. In Chinese philosophy, the concept of yin and yang ( is used to describe how seemingly opposing forces are bound together intertwined and interdependent in the Taoism (pronounced /ˈdaʊɪzəm/ or /ˈtaʊɪzəm/ also spelled '''Daoism''') refers to a variety of related Philosophical and Religious traditions The image, which is often misunderstood in the Western world as representing good (white) versus evil (black), is actually intended as a graphical representative of the complementary need for two abstract concepts of "maleness" (white) and "femaleness" (black). The term Western world, the West or the Occident ( Latin: occidens -sunset -west as distinct from the Orient) can have multiple meanings The symmetry of the symbol in this case is used not just to create a symbol that catches the attention of the eye, but to make a significant statement about the philosophical beliefs of the people and groups that use it. Also an important symmetrical religious symbol is the Shintoist "Torii" "The gate of the birds", usually the gate of the Shintoist temples called "Jinjas".

### Symmetry in Social Interactions

People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you". Peer relationships are based on symmetry, power relationships are based on asymmetry. [6]

### Symmetry in architecture

Another human endeavor in which the visual result plays a vital part in the overall result is architecture. The term architecture (from Greek αρχιτεκτονικήarchitektoniki) can be used to mean a process a profession or documentation Both in ancient times, the ability of a large structure to impress or even intimidate its viewers has often been a major part of its purpose, and the use of symmetry is an inescapable aspect of how to accomplish such goals.

Leaning Tower of Pisa

An interesting example of a broken symmetry in architecture is the Leaning Tower of Pisa, whose notoriety stems in no small part not for the intended symmetry of its design, but for the violation of that symmetry from the lean that developed while it was still under construction. Broken symmetry is a concept developed by Lee and Yang, used in Mathematics and Physics when an object breaks either Rotational symmetry The Leaning Tower of Pisa (Torre pendente di Pisa or simply The Tower of Pisa (it La Torre di Pisa is the Campanile, or freestanding bell tower of the Modern examples of architectures that make impressive or complex use of various symmetries include Australia's astonishing Sydney Opera House and Houston, Texas's simpler Astrodome. For a topic outline on this subject see List of basic Australia topics. The Sydney Opera House is located in Sydney New South Wales, Australia For the aeronautical use see Astrodome (aviation Reliant Astrodome, also known as the Houston Astrodome or simply the Astrodome

Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual floor plans, and down to the design of individual building elements such as intricately caved doors, stained glass windows, tile mosaics, friezes, stairwells, stair rails, and balustradess. A floor plan ( floorplan) in Architecture and Building engineering is a Diagram, usually to scale, of the relationships between rooms For the Blackford Oakes novel see Stained Glass (novel The term stained glass refers either to the material of coloured Glass or to the art Art History Mosaics of the 4th century BC are found in the Macedonian palace-city of Aegae, and they enriched the floors of Hellenistic In Architecture the frieze is the wide central section part of an Entablature and may be plain or &ndash in the Ionic or Corinthian order &ndash A baluster (according to OED derived through the French balustre, from Italian balaustro, from balaustra, "pomegranate flower" For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, Islamic buildings such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the general prohibition of Islam against using images or people or animals. For other meanings including people named 'Islam' see Islam (disambiguation.

Links related to symmetry in architecture include:

### Symmetry in pottery and metal vessels

Persian vessel (4th millennium B. C. )

Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. In Pottery, a potter's wheel is a machine used in the shaping of round ceramic wares As a minimum, pottery created using a wheel necessarily begins with full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point cultures from ancient times have tended to add further patterns that tend to exploit or in many cases reduce the original full rotational symmetry to a point where some specific visual objective is achieved. For example, Persian pottery dating from the fourth millennium B. The Persian Empire was a series of Iranian empires that ruled over the Iranian plateau, the original Persian homeland and beyond in Western Asia C. and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more complex and visually striking overall designs.

Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese, for example, used symmetrical patterns in their bronze castings as early as the 17th century B. The term Chinese people may refer to any of the following A person who resides in and holds citizenship of the People's Republic of China (including Hong C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.

### Symmetry in quilts

Kitchen Kaleidoscope Block

As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. A quilt is a type of Bedding — a bed covering composed of a quilt top a layer of batting, and a layer of fabric for backing generally combined using the technique

### Symmetry in carpets and rugs

Persian rug.

A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. A carpet is any loom-woven felted textile or grass floor covering American Navajo Indians used bold diagonals and rectangular motifs. The Navajo or Diné people (also spelled Navaho) of the Southwestern United States Many Oriental rugs have intricate reflected centers and borders that translate a pattern. An authentic oriental rug is a handmade Carpet that is either knotted with pile or woven without pile Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes. In art a motif is a repeated idea pattern image or theme Paisley designs are referred to as motifs

### Symmetry in music

Symmetry is of course not restricted to the visual arts. Its role in the history of music touches many aspects of the creation and perception of music. Music is an Art form in which the medium is Sound organized in Time.

#### Musical form

Symmetry has been used as a formal constraint by many composers, such as the arch form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney (or swell). The term musical form refers to two related concepts the type of composition (for example a musical work can have the form of a Symphony, a 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 WikipediaWikiProject Composers#Lead section --> Stephen Michael Reich (born October 3 Béla Viktor János Bartók (March 25 1881&ndashSeptember 26 1945 was a Hungarian Composer and Pianist, considered to be one of the greatest James Tenney ( August 10, 1934 - August 24, 2006) was an American Composer and influential music theorist. In classical music, Bach used the symmetry concepts of permutation and invariance; see (external link "Fugue No. 21," pdf or Shockwave).

#### Pitch structures

Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. In Music, a scale is a group of musical notes collected in ascending and descending order that provides material for or is used to conveniently represent part or all This article describes musical chords in traditional Western styles Tonality is a system of Music in which specific hierarchical pitch relationships are based on a key "center" or tonic. Pitch represents the perceived Fundamental frequency of a sound In Music theory, a diatonic scale (from the Greek διατονικος, meaning " through tones" also known as the heptatonia prima and In Music theory, a major chord ( is a chord having a root, a Major third, and a Perfect fifth. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. In Music, a whole tone scale is a scale in which each Note is separated from its neighbours by the interval of a Whole step. In general an augmented chord is any chord which contains an augmented interval. A seventh chord is a chord consisting of a triad plus a note forming an interval of a Seventh above the chord's root. Ambiguity (Am-big-u-i-ty is the property of being ambiguous, where a Word, term notation sign Symbol, Phrase, sentence, or any In Music theory, the term key is used in many different and sometimes contradictory ways A diatonic function, in tonal Music theory, is the specific recognized Roles of Notes or chords in relation to the key. However, composers such as Alban Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal centers. Alban Maria Johannes Berg (February 9 1885 &ndash December 24 1935 was an Austrian Composer. Béla Viktor János Bartók (March 25 1881&ndashSeptember 26 1945 was a Hungarian Composer and Pianist, considered to be one of the greatest George Perle (born May 6, 1915 in Bayonne New Jersey) is a Composer and music theorist. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical In Music theory, the term key is used in many different and sometimes contradictory ways Tonality is a system of Music in which specific hierarchical pitch relationships are based on a key "center" or tonic. The tonic is the first note of a musical scale in the tonal method of Musical composition.

Perle (1992) explains "C-E, D-F#, [and] Eb-G, are different instances of the same interval. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical . . the other kind of identity. . . has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:"

 D D# E F F# G G# D C# C B A# A G#

Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family (with C equal to 0).

 + 2 3 4 5 6 7 8 2 1 0 11 10 9 8 4 4 4 4 4 4 4

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin, Edgard Varèse, and the Vienna school. A chord progression (also chord sequence and harmonic progression or sequence) is a series of chords played in order Romantic Music is a Musicological term referring to a particular period theory compositional practice and canon in European music history from about 1815 to 1910 Alexander Nikolayevich Scriabin (Алекса́ндр Никола́евич Скря́бин Aleksandr Nikolaevič Skrjabin; sometimes transliterated as Skriabin WikipediaWikiProject Composers#Lead section --> Edgard Victor Achille Charles Varèse, whose name was also spelled Edgar Varèse At the same time, these progressions signal the end of tonality.

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)

#### Equivalency

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. In Music, a tone row or note row ( German: Reihe or Tonreihe) also series and set, refers to a non-repetitive In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e Invariance is a French journal edited by Jacques Camatte, published since 1968. Musical set theory provides concepts for categorizing musical objects and describing their relationships In Music theory, the word inversion has several meanings There are inverted chords, inverted melodies, inverted intervals, and See also Asymmetric rhythm. In Music, an additive rhythm is a Rhythm in which larger periods of time are constructed from sequences of smaller Rhythmic units added to the end of the

### Symmetry in other arts and crafts

Celtic knotwork

The concept of symmetry is applied to the design of objects of all shapes and sizes. Other examples include beadwork, furniture, sand paintings, knotwork, masks, musical instruments, and many other endeavors. Beadwork is the art or craft of attaching Beads to one another or to cloth using a needle and thread Furniture is the Mass noun for the movable objects which may support the human body (seating furniture and beds, provide storage or hold objects on horizontal KNOT (1450 AM) is a commercial Classic Country music Radio station in Prescott Arizona, broadcasting to the Flagstaff - Prescott A mask is an artefact normally worn on the face typically for protection concealment performance or amusement A musical instrument is a device constructed or modified for the purpose of making Music.

### Symmetry in aesthetics

The relationship of symmetry to aesthetics is complex. Symmetry, especially facial symmetry, is one of a number of Aesthetic traits including Averageness and Youthfulness, associated with Health Aesthetics or esthetics ( also spelled æsthetics) is commonly known as the study of sensory or sensori-emotional values sometimes called Certain simple symmetries, and in particular bilateral symmetry, seem to be deeply ingrained in the inherent perception by humans of the likely health or fitness of other living creatures, as can be seen by the simple experiment of distorting one side of the image of an attractive face and asking viewers to rate the attractiveness of the resulting image. "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Consequently, such symmetries that mimic biology tend to have an innate appeal that in turn drives a powerful tendency to create artifacts with similar symmetry. One only needs to imagine the difficulty in trying to market a highly asymmetrical car or truck to general automotive buyers to understand the power of biologically inspired symmetries such as bilateral symmetry. This article is about the semi-truck For the North American use of the word see Pickup truck.

Another more subtle appeal of symmetry is that of simplicity, which in turn has an implication of safety, security, and familiarity. A highly symmetrical room, for example, is unavoidably also a room in which anything out of place or potentially threatening can be identified easily and quickly. For example, people who have grown up in houses full of exact right angles and precisely identical artifacts can find their first experience in staying in a room with no exact right angles and no exactly identical artifacts to be highly disquieting. Symmetry thus can be a source of comfort not only as an indicator of biological health, but also of a safe and well-understood living environment.

Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Humans in particular have a powerful desire to exploit new opportunities or explore new possibilities, and an excessive degree of symmetry can convey a lack of such opportunities.

Yet another possibility is that when symmetries become too complex or too challenging, the human mind has a tendency to "tune them out" and perceive them in yet another fashion: as noise that conveys no useful information. is a one volume manga created by Tsutomu Nihei as a prequel to his ten-volume work Blame!.

Finally, perceptions and appreciation of symmetries are also dependent on cultural background. The far greater use of complex geometric symmetries in many Islamic cultures, for example, makes it more likely that people from such cultures will appreciate such art forms (or, conversely, to rebel against them). For other meanings including people named 'Islam' see Islam (disambiguation.

As in many human endeavors, the result of the confluence of many such factors is that effective use of symmetry in art and architecture is complex, intuitive, and highly dependent on the skills of the individuals who must weave and combine such factors within their own creative work. Along with texture, color, proportion, and other factors, symmetry is a powerful ingredient in any such synthesis; one only need to examine the Taj Mahal to powerful role that symmetry plays in determining the aesthetic appeal of an object. The Taj Mahal (tɑdʒ

A few examples of the more explicit use of symmetries in art can be found in the remarkable art of M. C. Escher, the creative design of the mathematical concept of a wallpaper group, and the many applications (both mathematical and real world) of tiling. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps

### Symmetry in games and puzzles

• See also symmetric games. In Game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them
• See sudoku. is a Logic -based number-placement Puzzle. The objective is to fill a 9×9 grid so that each column each row and each of the nine 3×3 boxes (also called blocks

Board games

### Symmetry in literature

See palindrome. A palindrome is a word phrase number or other sequence of units that can be read the same way in either direction (the adjustment of punctuation and spaces between words

### Moral symmetry

• Symmetry group
• Chirality
• Fixed points of isometry groups in Euclidean space - center of symmetry
• Spontaneous symmetry breaking
• Gödel, Escher, Bach
• M. C. Escher
• Wallpaper group
• Asymmetry
• Asymmetric rhythm
• Even and odd functions
• Symmetries of polyominoes
• Symmetries of polyiamonds
• Burnside's lemma
• Symmetry (biology)
• Spacetime symmetries
• Semimetric, which is sometimes translated as symmetric in Russian texts. Tit for tat is a highly effective strategy in Game theory for the Iterated prisoner's dilemma. In Social psychology, reciprocity refers to responding to a positive action with another positive action and responding to a negative action with another negative one Empathy is the capacity to recognize or understand another's state of mind or Emotion. Sympathy is a social affinity in which one person stands with another person closely understanding his or her feelings Reflective equilibrium is a state of Balance or coherence among a set of beliefs arrived at by a process of deliberative mutual adjustment among general principles and particular The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to A fixed point of an isometry group is a point that is a fixed point for every Isometry in the group In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state Gödel Escher Bach an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize -winning book by Douglas Hofstadter, described Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the Asymmetry is the absence of or a violation of a Symmetry. In organisms Due to how cells divide in Organisms asymmetry in organisms is In Music, an additive rhythm is a Rhythm in which larger periods of time are constructed from sequences of smaller Rhythmic units added to the end of the In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive In Recreational mathematics, a polyomino is a Polyform with the square as its base form A polyiamond (also polyamond or simply iamond) is a Polyform in which the base form is an Equilateral triangle. Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in Group "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Spacetime symmetries refers to aspects of Spacetime that can be described as exhibiting some form of Symmetry. In Topology, a semimetric space is a generalized Metric space in which the Triangle inequality is not required

## References

• Livio, Mario (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. New York: Simon & Schuster. ISBN 0-7432-5820-7.
• Perle, George (1990). George Perle (born May 6, 1915 in Bayonne New Jersey) is a Composer and music theorist. The Listening Composer, p. 112. California: University of California Press. ISBN 0-520-06991-9.
• Perle, George (1992). Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 81-96.
• Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
• Weyl, Hermann (1952). Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Symmetry. Princeton University Press. ISBN 0-691-02374-3.
• Hahn, Werner (1998). Symmetry As A Developmental Principle In Nature And Art World Scientific. ISBN 981-02-2363-3.