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Symmetry is a characteristic feature of geometrical shapes, systems, equations, and other real or conceptual objects —typically, in which one half of the object appears to be a reflection (i.e., a "mirror") of the other half. A sphere (< Greek σφαίÏα) is a perfectly symmetrical geometrical object. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
System (from the Latin (systēma), and this from the Greek (sustēma)) is an assemblage of entity/objects, real or abstract, comprising a whole with each and every component/element interacting or related to another one. ...
An equation is a mathematical statement, in symbols, that two things are the same. ...
In philosophy, an object is a thing, an entity, or a being. ...
One half is the fraction resulting from dividing one by two (½), or any number by its double; multiplication by one half is equivalent to division by two. ...
Look up reflection in Wiktionary, the free dictionary. ...
A mirror is a surface with good specular reflection that is smooth enough to form an image. ...
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 ω is a function of the form ω : X1 × … × Xk → 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. ...
A sphere rotating around its axis. ...
IT IS KNOWN AS MARK a lunitice insain int gw brain ...
Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...
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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 × X → X, 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: V → V acting on the set of functions x: V → W 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. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
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. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
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. ...
The adjective Boolean (sometimes boolean), coined in honor of George Boole, is used in many contexts: An evaluation that results in either TRUE or FALSE. A boolean value is a truth value, either true or false, often coded 1 and 0, respectively. ...
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, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
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 ∈ 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 Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, 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 In mathematics, a cylinder is a quadric, i. ...
Current (I) flowing through a wire produces a magnetic field (B) around the wire. ...
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. ...
Gyroscope. ...
The velocity of an object is simply its speed in a particular direction. ...
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. ...
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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:
[edit] 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 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, which can be written as...
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 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. ...
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Reflection symmetry See reflection symmetry. Figures with the axes of symmetry drawn in. ...
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Rotational symmetry See rotational symmetry. Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ...
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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. ...
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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.
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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 −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.
[edit] 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 has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ...
A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...
Screw axis symmetry In 3D, screw axis symmetry is invariance under a rotation about an axis combined with translation along that axis . The word screw has several different meanings: A screw is a type of threaded fastener for holding things together. ...
We can distinguish:
- there is invariance for every angle and a proportional translation distance, this applies e.g. for an infinite helix and double helix;
- the angle has no common divisor with 360°; the symmetry group is discrete, although the set of angles is not; it does not contain pure translations
- n-fold screw axis (angle of 360°/n)
See also space group. A helix (pl: helices), from the Greek word Îλικας/Îλιξ, is a twisted shape like a spring, screw or a spiral staircase. ...
The Double-Helix are an alien race in the Wing Commander science fiction series. ...
The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...
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Symmetry combinations See symmetry combinations. This articles discusses various symmetry combinations. ...
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Color With a color image one can associate a greyshade or black-and-white image. One way is to associate with each color a greyshade or either black or white. Alternatively, boundaries may be represented in black, and interior areas in white. When considering symmetry "ignoring colors" this tends to mean that dark colors become black and light colors white, or that boundaries become black. Sometimes there is only one meaningful conversion, in other cases the conversion has to be specified to avoid ambiguity (see e.g. the tetrakis square tiling). The new image may have more symmetry. Also colors may provide a special kind of symmetry, e.g. with corresponding points having opposite colors (including black and white), such as in the yin and yang symbol. Compare the modified symmetry model for vector fields, above. Image File history File links Yin_yang. ...
Image File history File links Yin_yang. ...
In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. ...
Taijitu, the traditional symbol representing the forces of Yin and Yang The concepts of yin and yang originate in ancient Chinese philosophy and metaphysics, which describes two primal opposing but complementary forces found in all things in the universe. ...
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Similarity vs. sameness Although two objects with great similarity appear the same, they must logically be different. For example, if one rotates an equilateral triangle around its center 120 degrees, it will appear the same as it was before the rotation to an observer. In theoretical euclidean geometry, such a rotation would be indistinguishable from its previous form. In reality however, each corner of any equilateral triangle composed of matter must be composed of separate molecules in separate locations. Therefore, symmetry in real physical objects is a matter of similarity instead of sameness. The difficulty for an intelligence to differentiate such a seemingly exact similarity is understandable. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...
Intelligence is the mental capacity to reason, plan, solve problems, think abstractly, comprehend ideas and language, and learn. ...
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More on symmetry in 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 – June 22, 1925) was a German mathematician. ...
An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
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. ...
A fractal, as conceived by Mandelbrot, has symmetry involving scaling. For example an equilateral triangle can be shrunk so that each of its sides are one third the length of the original's sides. These smaller triangles can be rotated and translated until they are adjacent and in the center of each of the larger triangle's lines. The smaller triangles can repeat the process, resulting in even smaller triangles on their sides. Fascinating intricate structures can be created by repeating such scaling symmetrical operations many times. It has been suggested that Fractal animation be merged into this article or section. ...
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 — a navigational aid which lists other pages...
For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
If a structure has a symmetry plane then for every part of the structure there are two possibilities:
- the part has itself a symmetry plane (the same plane)
- it has a mirror image counterpart
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Symmetry in mathematics - (main article: 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. ...
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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, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...
Symmetric binary logical connectives are "and" (∧, , 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. ...
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Generalization 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 Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...
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Symmetry in physics (see main article: symmetry in physics) This article or section does not cite its references or sources. ...
Symmetry in physics has been generalized to mean invariance (=unchange) under any kind of transformation. This 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. 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 — a navigational aid which lists other pages that might otherwise share...
Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between continuous symmetries and conservation laws. ...
In mathematics and theoretical physics, Wigners classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues. ...
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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. ...
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Symmetry in chemistry See Spectroscopy, Molecular orbital Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of spectra, that is, the dependence of physical quantities on frequency. ...
Electron atomic and molecular orbitals In quantum chemistry (electronic structure theory), the molecular electronic states, i. ...
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Symmetry in the arts and crafts You can find the use of symmetry across a wide variety of arts and crafts. [edit]
 The Parthenon on top of the Acropolis, Athens, Greece Architecture (from Latin, architectura and ultimately from Greek, αÏχιτεκτων, a master builder, from αÏχι- chief, leader and τεκτων, builder, carpenter) is the art and science of designing buildings and structures. ...
Wikipedia does not have an article with this exact name. ...
Symmetry has long been a predominant design element in architecture; prominent examples include the Leaning Tower of Pisa, Monticello, the Astrodome, the Sydney Opera House, Gothic church windows, and the Pantheon. Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, and ornamentation; many facades adhere to bilateral symmetry. The Tower of Pisa. ...
Thomas Jeffersons Monticello Monticello, located near Charlottesville, Virginia, was the estate of Thomas Jefferson, the third President of the United States. ...
The Reliant Astrodome, formerly just the Astrodome, is a domed sports stadium in Houston, Texas, and is part of the Reliant Park complex. ...
Internationally, the Sydney Opera House is the most recognised symbol of Sydney Sydney Opera House at Night The Sydney Opera House is located in Sydney, New South Wales, Australia. ...
See also Gothic art. ...
The Pantheon, Rome, in front of which stands the obelisk Macuteo, one of fourteen ancient Egyptian obelisks in Rome. ...
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. ...
Links:
- Williams: Symmetry in Architecture
- Aslaksen: Mathematics in Art and Architecture
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Image:Persian Pottery.jpg Unfired green ware pottery on a traditional drying rack at Conner Prairie living history museum. ...
The ancient Chinese used symmetrical patterns in their bronze castings since the 17th century B.C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design. Persian pottery dating from 6000 B.C. used symmetric zigzags, squares, and cross-hatchings.
Links:
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 An example of a patchwork quilt. ...
Quilt Block Pattern File links The following pages link to this file: Symmetry Categories: Images with unknown source ...
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.
Links:
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Image:Orientalrug.JPG A carpet is any loom-woven, felted textile or grass floor covering. ...
Look up Rug in Wiktionary, the free dictionary. ...
A long tradition of the use of symmetry in 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 most rugs use quadrilateral symmetry -- a motif reflected across both the horizontal and vertical axes.
Links:
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[edit] Music is an art, entertainment, or other human activity that involves organized and audible sounds and silence. ...
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 is used in two related ways: a generic type of composition such as the symphony or concerto the structure of a particular piece, how its parts are put together to make the whole; this too can be generic, such as binary form or sonata form Musical...
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. ...
Steve Reich Steve Reich (born Stephen Michael Reich, October 3, 1936) is an American composer. ...
Béla Bartók in 1927 For other uses, see Bartok (disambiguation). ...
James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ...
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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. ...
Fingering for a C-major trichord on a guitar in standard tuning (assuming all six strings are played). ...
Tonality is a system of writing music according to certain hierarchical pitch relationships around a center or tonic. ...
In music, pitch is the psychological correlate of the fundamental frequency of a note. ...
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. ...
Alban Maria Johannes Berg (February 9, 1885 – December 24, 1935) was an Austrian composer. ...
Béla Bartók in 1927 For other uses, see Bartok (disambiguation). ...
George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ...
In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ...
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 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, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ...
| 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 an order. ...
The era of Romantic music is defined as the period of European classical music that runs roughly from the early 1800s to the first decade of the 20th century, as well as music written according to the norms and styles of that period. ...
Gustav Mahler in 1909 Gustav Mahler (July 7, 1860 – May 18, 1911) was a Bohemian-Austrian composer and conductor. ...
Richard Wagner Wilhelm Richard Wagner (May 22, 1813 – February 13, 1883) was an influential German composer, conductor, music theorist, and essayist, primarily known for his operas (or music dramas as he later came to call them). ...
Alexander Nikolayevich Scriabin (Russian: ÐлекÑандр Ðиколаевич СкрÑбин; sometimes transliterated as Skryabin) (6 January 1872 – 27 April 1915) was a Russian composer and pianist. ...
This article is in need of improvement. ...
The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)
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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 things 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 — 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. ...
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 Celtic knotwork The concept of symmetry is applied to the design of objects of all shapes and sizes -- you can find it in the design of beadwork, furniture, sand paintings, knotwork, masks, and musical instruments (to name just a handful of examples). It has been suggested that Handicraft be merged into this article or section. ...
Celtic knotwork http://www. ...
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Symmetry does not by itself confer beauty to an object — many symmetrical designs are boring or overly challenging, and on the other hand preference for, or dislike of, exact symmetry is apparently dependent on cultural background. Along with texture, color, proportion, and other factors, symmetry does however play an important role in determining the aesthetic appeal of an object. See also M. C. Escher, wallpaper group, tiling. Kittens are often considered quite cute. ...
Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), 1935. ...
Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane crystallographic group) is a mathematical concept to classify repetitive designs on two-dimensional surfaces, such as walls, based on the symmetries in the pattern. ...
In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ...
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Symmetry in games and puzzles - See also symmetric games.
- See sudoku.
Puzzles A sudoku puzzle Sudoku ), also known as Number Place or Nanpure, is a logic-based placement puzzle. ...
Board Games [edit]
Symmetry in literature See palindrome. A palindrome is a word, phrase, number or other sequence of units (such as a strand of DNA) that has the property of reading the same in either direction (the adjustment of punctuation and spaces between words is generally permitted). ...
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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. [edit]
Moral symmetry [edit] 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. ...
Empathy is ones ability to recognize and understand the emotion of another. ...
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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 [edit] 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. ...
GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ...
Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), 1935. ...
Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane crystallographic group) is a mathematical concept to classify repetitive designs on two-dimensional surfaces, such as walls, based on the symmetries in the pattern. ...
In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well. ...
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. ...
A polyomino is a polyform with the square as its base form. ...
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, Pólyas 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. ...
References - Livio, Mario (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. New York: Simon & Schuster. ISBN 0-743-25820-7.
- 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.
[edit] George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ...
Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...
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