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Encyclopedia > Symmetry
Sphere symmetry group o.
Leonardo da Vinci's Vitruvian Man (1492) is often used as a representation of symmetry in the human body and, by extension, the natural universe.

Symmetry in common usage 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.[1] Image File history File links Sphere_symmetry_group_o. ... Image File history File links Sphere_symmetry_group_o. ... For other uses, see Sphere (disambiguation). ... Image File history File links Metadata Size of this preview: 441 Ã— 600 pixelsFull resolution (2258 Ã— 3070 pixel, file size: 5. ... Image File history File links Metadata Size of this preview: 441 Ã— 600 pixelsFull resolution (2258 Ã— 3070 pixel, file size: 5. ... â€œDa Vinciâ€ redirects here. ... Leonardo da Vincis Vitruvian Man (1492). ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... For other uses, see Geometry (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

Although the meanings are distinguishable, in some contexts, both meanings of "symmetry" are related and discussed in parallel.[2][3]

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

This article describes these notions of symmetry from three perspectives. The first is that of mathematics, in which symmetries are defined and categorized precisely. The second perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results of modern physics, including aspects of space and time. Finally, a third perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion. Look up time in Wiktionary, the free dictionary. ... This article is about the idea of space. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ... In the three-dimensional space, the possible moves of a rigid body are rotations and translations. ... For other uses, see Abstract It is a commonplace in philosophy that every thing or object is either abstract or concrete. ... Model may refer to more than one thing : For models in society, art, fashion, and cosmetics, see; role model model (person) supermodel figure drawing modeling section In science and technology, a model (abstract) is understood as an abstract or theoretical representation of a phenomenon,see; geologic modeling model (economics) model... For other uses, see Music (disambiguation). ... This article needs additional references or sources for verification. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... By the mid 20th century humans had achieved a mastery of technology sufficient to leave the surface of the Earth for the first time and explore space. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses of this term, see Spacetime (disambiguation). ... For other uses, see Humanities (disambiguation). ... This article is about the study of the past in human terms. ... This article is about building architecture. ... This article is about the philosophical concept of Art. ...

The opposite of symmetry is asymmetry. Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

## Symmetry in mathematics GA_googleFillSlot("encyclopedia_square");

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. 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). In logic and mathematics, an operation Ï‰ is a function of the form Ï‰ : X1 Ã— â€¦ Ã— Xk â†’ Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... This page includes English translations of several Latin phrases and abbreviations such as . ...

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 properties such as length. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... This article is about rotation as a movement of a physical body. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

### Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × XX, where the image of g in G and x in X is written as g·x. 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. 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 mathematics, a symmetry group describes all symmetries of objects. ... This picture illustrates how the hours on a clock form a group under modular addition. ... The symmetry group of an object (e. ...

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. 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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... This article or section does not cite its references or sources. ...

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. 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. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... For use in mathematics, see Boolean algebra (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. 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. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry. In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, given a lattice &#915; in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/&#915;, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

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. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. 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. A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... This gyroscope remains upright while spinning due to its angular momentum. ... This article is about velocity in physics. ...

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 share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

### 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 is the symmetry group associated with Euclidean geometry. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

Examples:

• G is the group of affine transformations with a matrix A with determinant 1 or -1, i.e. the transformation which preserve area; this adds e.g. oblique reflection symmetry.
• G is the group of all bijective affine transformations
• More generally, an involution defines a symmetry with respect to that involution.

Several equivalence relations in mathematics are called similarity. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, a dilation is a function f from a metric space into itself that satisfies the identity where d(x,y) is the distance from x to y and r is some positive real number. ... A self-similar object is exactly or approximately similar to a part of itself. ... Figures with the axes of symmetry drawn in. ... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

### Directional symmetry

See directional symmetry. The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

### Reflection symmetry

See reflection symmetry. Figures with the axes of symmetry drawn in. ...

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.

It is the most common type of symmetry[citation needed]. 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). A mirror image is a mirror based duplicate of a single image. ...

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. A circle has infinitely many axes of symmetry, for the same reason. For other uses, see Square. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

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. ... A triangle. ... This article is about the geometric shape. ... A kite showing its equal sides and its inscribed circle. ... This article is about the geometric figure. ...

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. The fundamental domain is a half-plane or half-space. In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ...

Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane. Illustration of the different types of symmetry of Life Forms On Earth. ... Diagram showing mid-sagittal, coronal and transverse planes. ...

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). Look up Parity in Wiktionary, the free dictionary Parity is a concept of equality of status or functional equivalence. ...

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

Look up isometric in Wiktionary, the free dictionary. ... For uses in mathematics see: Affine transformation Affine combination Affine geometry Affine space Affine group Affine representation Affine variety Affine scheme Affine cipher This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ...

### Rotational symmetry

See rotational symmetry. The triskelion appearing on the Isle of Man flag. ...

Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m) (see Euclidean group). In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

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. For m=3 this is the rotation group. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

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. See also rotational invariance. Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... This article or section does not cite its references or sources. ...

### Translational symmetry

See main article translational symmetry. A translation slides an object by a vector a: Ta(p) = p + a. ...

Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(p) = p + a In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

### 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. It implies translational symmetry with twice the translation vector. Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

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. As symmetry groups with regard to a roto-reflection we can distinguish: In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to &#8722;x). ...

• 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.
• 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 mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... InVersion are a heavy metal band from Essex who came together with the aim to blend the melody of old school metal with the aggression and rhythm of more modern bands. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

### Helical symmetry

A drill bit with helical symmetry.
See also screw axis.

Helical symmetry is the kind of symmetry seen in such everyday objects as springs, Slinky toys, drill bits, and augers. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis). 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). 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°. Image File history File links 400px-Drillbit. ... Image File history File links 400px-Drillbit. ... // 31 screw axis in crystal structure of tellurium In crystallography, a screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged. ... A helix (pl: helices), from the Greek word Î­Î»Î¹ÎºÎ±Ï‚/Î­Î»Î¹Î¾, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ... For other uses, see Spring. ... Metal Slinky Rainbow-colored plastic Slinky A Slinky, or Lazy-Spring, is a coil-shaped toy invented by mechanical engineer Richard James in Philadelphia, Pennsylvania. ... Drill bits are the cutters of drill tools. ... Study of a man using an auger, for The Seven Sorrows of the Virgin, Albrecht DÃ¼rer, ca 1496 An auger is a device for moving material or liquid by means of a rotating helical flighting. ... // 31 screw axis in crystal structure of tellurium In crystallography, a screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency &#969; (also called angular speed) is a scalar measure of rotation rate. ...

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-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. 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. 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. Such symmetries are created by using a non-repeating point group in two dimensions. DNA is an example of this type of non-repeating helical symmetry.

A helix (pl: helices), from the Greek word Î­Î»Î¹ÎºÎ±Ï‚/Î­Î»Î¹Î¾, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ... Cross section may refer to the following In geometry, Cross section is the intersection of a 3-dimensional body with a plane. ... For other uses, see Spring. ... Metal Slinky Rainbow-colored plastic Slinky A Slinky, or Lazy-Spring, is a coil-shaped toy invented by mechanical engineer Richard James in Philadelphia, Pennsylvania. ... Drill bits are cutting tools used to create cylindrical holes. ... Study of a man using an auger, for The Seven Sorrows of the Virgin, Albrecht DÃ¼rer, ca 1496 An auger is a device for moving material or liquid by means of a rotating helical flighting. ... The Double-Helix are an alien race in the Wing Commander science fiction series. ... In mathematics and physics, the radian is a unit of angle measure. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... The structure of part of a DNA double helix Deoxyribonucleic acid, or DNA, is a nucleic acid molecule that contains the genetic instructions used in the development and functioning of all known living organisms. ...

### 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. 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. Galileo can refer to: Galileo Galilei, astronomer, philosopher, and physicist (1564 - 1642) the Galileo spacecraft, a NASA space probe that visited Jupiter and its moons the Galileo positioning system Life of Galileo, a play by Bertolt Brecht Galileo (1975) - screen adaptation of the play Life of Galileo by Bertolt Brecht... Genera and Species Loxodonta Loxodonta cyclotis Loxodonta africana Elephas Elephas maximus Elephas antiquus â€  Elephas beyeri â€  Elephas celebensis â€  Elephas cypriotes â€  Elephas ekorensis â€  Elephas falconeri â€  Elephas iolensis â€  Elephas planifrons â€  Elephas platycephalus â€  Elephas recki â€  Stegodon â€  Mammuthus â€  Elephantidae (the elephants) is a family of pachyderm, and the only remaining family in the order Proboscidea... This article is about the animal. ...

A more subtle form of scale symmetry is demonstrated by fractals. 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. 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 branching of trees, which enables children to use small twigs as stand-ins for full trees in dioramas, is another example. The boundary of the Mandelbrot set is a famous example of a fractal. ... Mandelbrot set, popularized by Benoît Mandelbrot Mandelbrot, (ger; almond-bread ), may refer to: Benoît Mandelbrot, a French mathematician largely responsible for later interest in fractal geometry Mandelbrot set, a fractal popularized by Benoît Mandelbrot This is a disambiguation page &#8212; a navigational aid which lists other pages... Magnification is the process of enlarging something only in appearance, not physical size. ... For other uses, see Coast (disambiguation). ... A diorama is any of the two display devices mentioned below. ...

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. 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. Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane that forms a fractal. ... CG or Cg may stand for: computer graphics center of gravity CG artwork, digitally made artwork, digital effects, or simply hand drawn art saved on a computer Cg programming language, developed by NVIDIA Chappe et Gessalin, a French automobile maker Character generator, broadcast graphic systems Coast guard Confused Gamer Conceptual... This article does not cite any references or sources. ...

### Symmetry combinations

See symmetry combinations. This articles discusses various symmetry combinations. ...

## Symmetry in science and technology

### 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 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. See Noether's theorem (which, as a gross oversimplification, states that for every 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. This article or section does not cite its references or sources. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... In mathematics and theoretical physics, Wigners classification is a classification of the nonnegative energy irreducible unitary representations of the PoincarÃ© group, which have sharp mass eigenvalues. ...

### 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. 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; she will immediately recognize that the object has been rotated by looking for details such as crystals or minor deformities. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A 1879 Carl Zeiss Jena Optical microscope. ... An electron microscope is a type of microscope that uses electrons as a way to illuminate and create an image of a specimen. ... For other uses, see Crystal (disambiguation). ...

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. 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. In philosophy, physics, and other fields, a thought experiment (from the German Gedankenexperiment) is an attempt to solve a problem using the power of human imagination. ... This article or section does not cite its references or sources. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ...

#### 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. Fig. ... For other uses, see Electron (disambiguation). ... For other uses, see Proton (disambiguation). ... For other uses, see Light (disambiguation). ... Properties For alternative meanings see atom (disambiguation). ...

Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. 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. 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. For other uses, see Electron (disambiguation). ... In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ... For other uses, see Observation (disambiguation). ...

#### 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 3.4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture.) This article is about the physicist. ... The Feynman Lectures on Physics, by Richard Feynman, is perhaps his most accessible technical work for anyone with an interest in physics and today is considered to be the classic introduction to modern physics, including lectures on mathematics, electromagnetism, Newtonian physics, quantum physics, and even the relation of physics to...

"... 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. Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Surface waves in water This article is about waves in the most general scientific sense. ...

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. Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... Fig. ...

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'). 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). By now it has been applied in numerous forms, as kind of standard attack on problems. Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. ... New math is a term referring to a brief dramatic change in the way mathematics was taught in American grade schools during the 1960s. ...

### Symmetry in mathematics

An example of a mathematical expression exhibiting symmetry is a²c + 3ab + b²c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication. Symmetry in mathematics occurs not only in geometry, but also in other branches of mathematics. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

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

See also symmetric function, duality (mathematics) In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ... In mathematics, duality has numerous meanings. ...

### Symmetry in logic

A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, “is the same age as” is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...

Symmetric binary logical connectives are "and" (∧, $land$, or &), "or" (∨), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or"). In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic 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 (e. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...

Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups. This article or section is in need of attention from an expert on the subject. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here &#916; is the comultiplication of the bialgebra, &#8711; its multiplication, &#951; its unit and &#949; its counit. ...

### Symmetry in biology

See symmetry (biology) and facial symmetry. The elaborate patterns on the wings of butterflies are one example of biological symmetry. ... Facial symmetry is one of a number of traits associated with health, physical attractiveness and beauty of a person or animal. ...

### Symmetry in chemistry

Main article: molecular symmetry

Symmetry is important to chemistry because it explains observations in spectroscopy, quantum chemistry and crystallography. It draws heavily on group theory. Molecular symmetry in chemistry describes symmetry in molecules and the classification of molecules in groups based on symmetry. ... For other uses, see Chemistry (disambiguation). ... Animation of the dispersion of light as it travels through a triangular prism. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Group theory is that branch of mathematics concerned with the study of groups. ...

### Symmetry in telecommunications

Some telecommunications services (specifically data products) may be referred to as symmetrical or asymmetrical. This refers to the bandwidth allocated for data sent and received. Most internet services used by residential customers are asymmetrical: the data sent to the server normally is far less than that returned by the server.

## 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 2. Islamic, Buddhist, Shinto 3. 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. 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. 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Topics in Christianity Movements Â· Denominations Â· Other religions Ecumenism Â· Preaching Â· Prayer Music Â· Liturgy Â· Calendar Symbols Â· Art Â· Criticism Important figures Apostle Paul Â· Church Fathers Constantine Â· Athanasius Â· Augustine Anselm Â· Aquinas Â· Palamas Â· Luther Calvin Â· Wesley Arius Â· Marcion of Sinope Archbishop of Canterbury Â· Catholic Pope Coptic Pope Â· Ecumenical Patriarch Christianity Portal This box:      Christianity is... This article or section does not cite its references or sources. ... Hinduism (known as in modern Indian languages[1]) is a religious tradition[2] that originated in the Indian subcontinent. ... For people named Islam, see Islam (name). ... A statue of the Sakyamuni Buddha in Tawang Gompa, India. ... Shinto ) is the native religion of Japan and was once its state religion. ... Sikhism (IPA: or ; Punjabi: , , IPA: ) is a religion that began in fifteenth century Northern India with the teachings of Nanak and nine successive human gurus. ... This article is about the generally-recognized global religious community. ... Jain and Jaina redirect here. ... The triskelion appearing on the Isle of Man flag. ... This article or section does not cite its references or sources. ... This article is about a Jewish symbol. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... Taoism (or Daoism) is the English name referring to a variety of related Chinese philosophical traditions and concepts. ... A commonly used version of the Taijitu The Taijitu of Zhou Dun-yi. ... In biology, bilateral symmetry is a characteristic of multicellular organisms, particularly animals. ... Topics in Christianity Movements Â· Denominations Â· Other religions Ecumenism Â· Preaching Â· Prayer Music Â· Liturgy Â· Calendar Symbols Â· Art Â· Criticism Important figures Apostle Paul Â· Church Fathers Constantine Â· Athanasius Â· Augustine Anselm Â· Aquinas Â· Palamas Â· Luther Calvin Â· Wesley Arius Â· Marcion of Sinope Archbishop of Canterbury Â· Catholic Pope Coptic Pope Â· Ecumenical Patriarch Christianity Portal This box:      Christianity is... Also known as the Latin cross or crux ordinaria. ... Sikhism (IPA: or ; Punjabi: , , IPA: ) is a religion that began in fifteenth century Northern India with the teachings of Nanak and nine successive human gurus. ... The Khanda Sikh Khanda on Stamp designed by Stacey Zabolotney Issued By Canada Post in November 2000 . ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... This article is about the symbol. ... For people named Islam, see Islam (name). ... Sunni Islam (Arabic &#1587;&#1606;&#1617;&#1577;) is the largest denomination of Islam. ...

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. 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 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". A commonly used version of the Taijitu The Taijitu of Zhou Dun-yi. ... Taoism (or Daoism) is the English name referring to a variety of related Chinese philosophical traditions and concepts. ... Occident redirects here. ...

### 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. [7]

### Symmetry in architecture

Another human endeavor in which the visual result plays a vital part in the overall result is architecture. 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. This article is about building architecture. ...

Just a few examples of ancient examples of architectures that made powerful use of symmetry to impress those around them included the Egyptian Pyramids, the Greek Parthenon, and the first and second Temple of Jerusalem, China's Forbidden City, Cambodia's Angkor Wat complex, and the many temples and pyramids of ancient Pre-Columbian civilizations. More recent historical examples of architectures emphasizing symmetries include Gothic architecture cathedrals, and American President Thomas Jefferson's Monticello home. India's unparalleled Taj Mahal is in a category by itself, as it may arguably be one of the most impressive and beautiful uses of symmetry in architecture that the world has ever seen. This is about the polyhedron. ... For other uses, see Parthenon (disambiguation). ... The Jerusalem Temple (Hebrew: beit ha-mikdash) was the center of Israelite and Jewish worship, primarily for the offering of sacrifices known as the korbanot. ... For other uses, see Forbidden City (disambiguation). ... Aerial view of Angkor Wat The main entrance to the temple proper, seen from the eastern end of the Naga causeway Angkor Wat (or Angkor Vat) is a temple at Angkor, Cambodia, built for King Suryavarman II in the early 12th century as his state temple and capital city. ... The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the Americas continent. ... The western facade of Reims Cathedral, France. ... Thomas Jefferson (13 April 1743 N.S.â€“4 July 1826) was the third President of the United States (1801â€“09), the principal author of the Declaration of Independence (1776), and one of the most influential Founding Fathers for his promotion of the ideals of Republicanism in the United States. ... This is about the Jefferson residence. ... Taj Mahal Location of the Taj Mahal within India The Taj Mahal (Devanagari: à¤¤à¤¾à¤œ à¤®à¤¹à¤², Nastaliq: ØªØ§Ø¬ Ù…Ø­Ù„) is a mausoleum located in Agra, India. ...

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. 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. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... Broken symmetry is a concept used in mathematics and physics when an object breaks either rotational symmetry or translational symmetry. ... The Leaning Tower of Pisa (Italian: ) or simply The Tower of Pisa (La Torre di Pisa) is the campanile, or freestanding bell tower, of the cathedral of the Italian city of Pisa. ... The Sydney Opera House is located in Sydney, New South Wales, Australia. ... Houston redirects here. ... The Reliant Astrodome, formerly just the Astrodome, is a domed sports stadium in Houston, Texas, and is part of the Reliant Park complex. ...

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. 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. Floor plan (floorplan, floor-plan) in its original meaning is an architecture term, a diagram of a room, a building, or a level (floor) of a building as if seen from the above (i. ... Strictly speaking, stained glass is glass that has been painted with silver stain and then fired. ... This article is about a decorative art. ... Frieze of the Tower of the Winds. ... A page of fanciful balusters from A Handbook of Ornament, Franz S. Meyer, 1898 A baluster (through the French balustre, from Italian balaustro, from balaustra, pomegranate flower [from a resemblance to the post], from Lat. ... For people named Islam, see Islam (name). ...

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. 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.C. and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more complex and visually striking overall designs. Image File history File links Download high resolution version (764x918, 385 KB)Photo by Zereshk. ... Image File history File links Download high resolution version (764x918, 385 KB)Photo by Zereshk. ... The potters wheel is a horizontal wheel or turntable used in the making of many types of pottery. ... For other uses of this term see: Persia (disambiguation) The Persian Empire is the name used to refer to a number of historic dynasties that have ruled the country of Persia (Iran). ...

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.C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.

Links:

• The Metropolitan Museum of Art - Islamic Art

### 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. Image File history File links Kitchen_kaleid. ... Image File history File links Kitchen_kaleid. ... A quilt is a type of puppy with long fluffy ears. ...

Links:

### 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. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes. Image File history File linksMetadata Farsh1. ... Image File history File linksMetadata Farsh1. ... For other uses, see Carpet (disambiguation). ... Look up Rug in Wiktionary, the free dictionary. ... The Navajo (also Navaho) people of the southwestern United States call themselves the DinÃ© (pronounced ), which roughly means the people. They speak the Navajo language, and many are members of the Navajo Nation, an independent government structure which manages the Navajo reservation in the Four Cs area of the United... An authentic oriental rug is a handmade carpet that is either knotted with pile or woven without pile. ... Look up motif in Wiktionary, the free dictionary. ...

Links:

• Mallet: Tribal Oriental Rugs

### Symmetry in music

Symmetry is of course not restricted to the visual arts. Its role in the history of music touches many aspect of the creation and perception of music. For other uses, see Music (disambiguation). ...

#### 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). In classical music, Bach used the symmetry concepts of permutation and invariance; see (external link "Fugue No. 21," pdf or Shockwave). 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 concerto, or other generic type -- see Multi-movement forms below) the structure of a particular piece (for example, a piece can be written in... In music, arch form is a sectional way of structuring a piece of music based on the repetition, in reverse order, of all or most musical sections such that the overall form is symmetrical, most often around a central movement. ... Stephen Michael Reich (born October 3, 1936) is an American composer. ... Bartok redirects here. ... James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ...

#### 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. 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. 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. In music, a scale is a set of musical notes that provides material for part or all of a musical work. ... Typical fingering for a second inversion C major chord on a guitar. ... Tonality is a system of writing music according to certain hierarchical pitch relationships around a key center or tonic. ... Pitch is the perceived fundamental frequency of a sound. ... In music theory, a diatonic scale (from the Greek diatonikos, to stretch out; also known as the heptatonia prima; set form 7-35) is a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated. ... Generally speaking, a major chord is any chord which has a major third above its root, as opposed to a minor chord which has a minor third. ... In music, a whole tone scale (set form 6-35, 02468t) is a scale in which each note is separated from its neighbors 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 chords root. ... - Emo Philips A word, phrase, sentence, or other communication is called ambiguous if it can be reasonably interpreted in more than one way. ... In music theory, the key identifies the tonic triad, the chord, major or minor, which represents the final point of rest for a piece, or the focal point of a section. ... See also: function and functional. ... Bust of Alban Berg at Schiefling, Carinthia, Austria Alban Maria Johannes Berg (February 9, 1885 â€“ December 24, 1935) was an Austrian composer. ... Bartok redirects here. ... George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ... In music theory, the term interval describes the difference in pitch between two notes. ... In Music theory, the key is the tonal center of a piece. ... Tonality is a system of writing music according to certain hierarchical pitch relationships around a key center or tonic. ... The tonic is the first note of a musical scale, and in the tonal method of music composition it is extremely important. ...

Perle (1992) explains "C-E, D-F#, [and] Eb-G, are different instances of the same interval...the other kind of identity. ..has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:" In music theory, the term interval describes the difference in pitch between two notes. ...

 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. 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. At the same time, these progressions signal the end of tonality. In music, an enharmonic is a note which is the equivalent of some other note, but spelled differently. ... A chord progression (also chord sequence and harmonic progression or sequence), as its name implies, is a series of chords played in order. ... The era of Romantic music is defined as the period of European classical music that runs roughly from 1820 to 1900, as well as music written according to the norms and styles of that period. ... â€œMahlerâ€ redirects here. ... Richard Wagner Wilhelm Richard Wagner (22 May 1813 â€“ 13 February 1883) was a German composer, conductor, music theorist, and essayist, primarily known for his operas (or music dramas as they were later called). ... Alexander Nikolayevich Scriabin Alexander Nikolayevich Scriabin (Russian: ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ€ ÐÐ¸ÐºÐ¾Ð»Ð°ÐµÐ²Ð¸Ñ‡ Ð¡ÐºÑ€ÑÐ±Ð¸Ð½, Aleksandr NikolajeviÄ Skriabin; sometimes transliterated as Skryabin or Scriabine (6 January 1872 [O.S. 26 December 1871]â€”27 April 1915) was a Russian composer and pianist. ... Edgard Victor Achille Charles VarÃ¨se (December 22, 1883 â€“ November 6, 1965) was a French-born composer. ...

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. See also Asymmetric rhythm. In music, a tone row or note row is a permutation, an arrangement or ordering, of the twelve notes of the chromatic scale. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share... Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... In music theory, the word inversion has several meanings. ... Additive rhythms are larger periods of time constructed from sequences of smaller rhythmic units added to the end of the previous unit. ...

### 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. Celtic knotwork http://www. ... Beadwork is the art or craft of attaching beads to one another or to cloth using a needle and thread. ... For the UK band, see Furniture (band). ... KNOT is a commercial Classic Country music radio station in Prescott, Arizona, broadcasting to the Flagstaff-Prescott, Arizona area on 1450 AM. Query the FCCs AM station database for KNOT Radio Locator Information on KNOT AM radio stations in the Flagstaff-Prescott, Arizona market (Arbitron #151) By frequency: By... This article is about masks fitted on the face as an article of clothing or equipment. ... A musical instrument is a device that has been constructed or modified with the purpose of making music. ...

### Symmetry in aesthetics

The relationship of symmetry to aesthetics is complex. 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. 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. In evolutionary psychology, symmetry especially facial symmetry is one of a number of traits, including averageness and youthfulness, associated with health, physical attractiveness and beauty of a person or animal. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... In biology, bilateral symmetry is a characteristic of multicellular organisms, particularly animals. ... This article or section does not cite its references or sources. ... For other uses, see Truck (disambiguation). ...

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. People who have, for example, 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. This article is about noise as in sound. ...

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 people named Islam, see Islam (name). ...

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. Taj Mahal Location of the Taj Mahal within India The Taj Mahal (Devanagari: à¤¤à¤¾à¤œ à¤®à¤¹à¤², Nastaliq: ØªØ§Ø¬ Ù…Ø­Ù„) is a mausoleum located in Agra, India. ...

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 (June 17, 1898 â€“ March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ... A tessellated plane seen in street pavement. ...

### Symmetry in games and puzzles

• See also symmetric games.

Board games This article is about the logic puzzle. ...

### Symmetry in literature

See palindrome. For the movie, see Palindromes (film). ...

### Moral symmetry

Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ... In social psychology, reciprocity refers to in-kind positive or negative responses of individuals towards the actions of others. ... The term Golden Rule may refer to any of the following Wikipedia articles: The Golden Rule - in ethics, religion and philosophy. ... Not to be confused with Pity, Sympathy, or Compassion. ... ... 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 judgments. ...

## See also

The symmetry group of an object (e. ... 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 its mirror image by rotations and translations alone. ... A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... GÃ¶del, Escher, Bach: an Eternal Golden Braid: A metaphorical fugue on minds and machines in the spirit of Lewis Carroll (commonly GEB) is a Pulitzer Prize (1980)-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... Maurits Cornelis Escher (June 17, 1898 â€“ March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Additive rhythms are larger periods of time constructed from sequences of smaller rhythmic units added to the end of the previous unit. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ... Dynamic symmetry is a proportioning system originated from the classical Greek period. ... The 35 possible hexominoes. ... A polyiamond is a counterpart to a polyomino where the polygon used as the building block is an equilateral triangle rather than a square. ... Burnsides lemma, sometimes also called Burnsides counting theorem, PÃ³lyas formula, the Cauchy-Frobenius lemma or the Orbit-Counting Theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ... The elaborate patterns on the wings of butterflies are one example of biological symmetry. ... The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... In mathematics, a semimetric space generalizes the concept of a metric space by not requiring the condition of satisfying the triangle inequality. ...

## 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). 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). 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.
• Symmetry: Culture and Science, published by Symmetrion, Budapest. ISSN 0865-4824.
• Darvas, György (2007). Symmetry, Basel-Berlin-Boston: Birkhäuser Verlag, xi + 508 p.

George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ... Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ...

## Notes

1. ^ Weyl, Hermann (1989). Symmetry. Princeton University Press. ISBN 0691023743.
2. ^ (Wey 1989)
3. ^ For example, Aristotle ascribed spherical shape to the heavenly bodies, attributing this formally-defined geometric measure of symmetry to the natural order and perfection of the cosmos.
4. ^ For example, operations such as moving across a regularly patterned tile floor or rotating an eight-sided vase, or complex transformations of equations or in the way music is played.
5. ^ See e.g., Mainzer, Klaus (2005). Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science. World Scientific. ISBN 9812561927.
6. ^ Symmetric objects can be material, such as a person, crystal, quilt, floor tiles, or molecule, or it can be an abstract structure such as a mathematical equation or a series of tones (music).
7. ^ Emotional Competency Entry describing Symmetry

For other uses, see Aristotle (disambiguation). ... Chinese vase A vase with a sunflower pattern A modern designed vase The vase is an open container, often used to hold cut flowers. ... For other uses, see Crystal (disambiguation). ... A quilt is a type of puppy with long fluffy ears. ... Mission, or barrel, roof tiles A tile is a manufactured piece of hard-wearing material such as ceramic, stone, porcelain, metal or even glass. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... For other uses, see Abstract It is a commonplace in philosophy that every thing or object is either abstract or concrete. ... An equation is a mathematical statement, in symbols, that two things are the same. ... For other uses, see Music (disambiguation). ...

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