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Symmetry

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Sphere symmetry group o.
   Sphere symmetry group o.

   Symmetry means "constancy", i.e. if something retains a certain feature
   even after we change a way of looking at it, then it is symmetric.

   Thus things can be symmetric
     * over time (if they do not change over time)
     * over space (if they look the same even from different points of
       view)
     * over size (if it looks the same even if we zoom in or out).

   The symmetric object 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). The nature of the
   change can be similarly diverse, ranging from such simple and intuitive
   operations as moving across a regularly patterned tile floor or
   rotating an eight-sided vase, to complex transformations of equations
   or in the way music is played.

   This article describes symmetry from three perspectives. The first is
   that of mathematics, in which symmetries are defined and categorized
   precisely. The second perspective the application of symmetry concepts
   to science and technology. For example, symmetries turn out to underlie
   some of the most profound results of modern physics, including aspects
   of space and time. Finally, a third perspective looks at the rich and
   varied use of symmetry in history, architecture, art, and religion.

   The opposite of symmetry is asymmetry.

Symmetry in mathematics

   In formal terms, we say that an object is symmetric with respect to a
   given mathematical operation, if, when applied to the object, this
   operation does not change the object or its appearance. 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).

   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.

Mathematical model for symmetry

Non-isometric symmetry

   As mentioned above, G (the symmetry group of the space itself) may
   differ from the Euclidean group, the group of isometries.

   Examples:
     * G is the group of similarity transformations, i.e. affine
       transformations with a matrix A that is a scalar times an
       orthogonal matrix. Thus dilations are added, self-similarity is
       considered a symmetry

     * G is the group of affine transformations with a matrix A with
       determinant 1 or -1, i.e. the transformation which preserve area;
       this adds e.g. oblique reflection symmetry.

     * G is the group of all bijective affine transformations

     * In inversive geometry, G includes circle reflections, etc.

     * More generally, an involution defines a symmetry with respect to
       that involution.

Directional symmetry

   See directional symmetry.

Reflection symmetry

   See reflection symmetry.

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

   It is the most common type of symmetry. In 2D there is an axis of
   symmetry, in 3D a plane of symmetry. An object or figure which is
   indistinguishable from its transformed image is called mirror symmetric
   (see mirror image).

   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.

   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.

   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.

   Bilateria (bilateral animals, including humans) are more or less
   symmetric with respect to the sagittal plane.

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

   For more general types of reflection there are corresponding more
   general types of reflection symmetry. Examples:
     * with respect to a non- isometric affine involution (an oblique
       reflection in a line, plane, etc).

     * with respect to circle inversion

Rotational symmetry

   See rotational symmetry.

   Rotational symmetry is symmetry with respect to some or all rotations
   in m-dimensional Euclidean space. Rotations are direct isometries,
   i.e., isometries preserving orientation. Therefore a symmetry group of
   rotational symmetry is a subgroup of E+(m) (see Euclidean group).

   Symmetry 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 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.

Translational symmetry

   See main article translational symmetry.

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

Glide reflection symmetry

   A glide reflection symmetry (in 3D: a glide plane symmetry) means that
   a reflection in a line or plane combined with a translation along the
   line / in the plane, results in the same object. It implies
   translational symmetry with twice the translation vector.

   The symmetry group is isomorphic with Z.

Rotoreflection symmetry

   In 3D, rotoreflection or improper rotation in the strict sense is
   rotation about an axis, combined with reflection in a plane
   perpendicular to that axis. As symmetry groups with regard to a
   roto-reflection we can distinguish:
     * the angle has no common divisor with 360°, the symmetry group is
       not discrete
     * 2n-fold rotoreflection (angle of 180°/n) with symmetry group S[2n]
       of order 2n (not to be confused with symmetric groups, for which
       the same notation is used; abstract group C[2n]); 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.
     * C[nh] (angle of 360°/n); for odd n this is generated by a single
       symmetry, and the abstract group is C[2n], for even n this is not a
       basic symmetry but a combination.See also point groups in three
       dimensions.

Helical symmetry

   A drill bit with helical symmetry.
   A drill bit with helical symmetry.

   Helical symmetry is the kind of symmetry seen in such everyday objects
   as springs, Slinky toys, drill bits, and augers. It can be thought of
   as rotational symmetry along with translation along the axis of
   rotation. The concept of helical symmetry can be visualized as the
   tracing in three-dimensional space that results from rotating an object
   at an even 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°.

   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. 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.

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.

   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.

   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.

Symmetry combinations

   See symmetry combinations.

Symmetry in science and technology

Generalisation of the concept of symmetry

     * Definitions of symmetry and its related notions

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.

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 aluminium equilateral
   triangle 120 degrees around its centre, 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.

   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.

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.

   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.

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.)

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

   The word " interferes" in this context is a quick way of saying that
   such objects fall under the rules of quantum mechanics, in which they
   behave more like waves that interfere than like everyday large objects.

   In 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.

   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.

Symmetry in mathematics

   An example of a mathematical expression exhibiting symmetry is a^2c +
   3ab + b^2c. If a and b are exchanged, the expression remains unchanged
   due to the commutativity of addition and multiplication.

   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)

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.

   Symmetric binary logical connectives are " and" (∧, \land , or &), "
   or" (∨), " biconditional" ( iff) (↔), NAND ("not-and"), XOR
   ("not-biconditional"), and NOR ("not-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.

   Physicists have come up with other directions of generalization, such
   as supersymmetry and quantum groups.

Symmetry in biology

   See symmetry (biology) and facial symmetry.

Symmetry in chemistry

   Symmetry is important to chemistry because it explains observations in
   spectroscopy, quantum chemistry and crystallography. It draws heavily
   on group theory.

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 do not play a profound role at all.
   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
   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.

   The ancient Taijitu image of Taoism is a particularly fascinating use
   of symmetry around a central point, combined with black-and-white
   inversion of colour 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".

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.

   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, Kampuchea's (Cambodia's) Angkor
   Wat complex, and the 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.
   Leaning Tower of Pisa
   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.

   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.

   Links related to symmetry in architecture include:
     * Williams: Symmetry in Architecture
     * Aslaksen: Mathematics in Art and Architecture

Symmetry in pottery and metal vessels

   Persian vessel (4th millennium B.C.)
   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.

   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:
     * Chinavoc: The Art of Chinese Bronzes

     * Grant: Iranian Pottery in the Oriental Institute

     * The Metropolitan Museum of Art - Islamic Art

Symmetry in quilts

   Kitchen Kaleidoscope Block
   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.

   Links:
     * Quate: Exploring Geometry Through Quilts

     * Quilt Geometry

Symmetry in carpets and rugs

   Navajo blanket
   Navajo blanket
   Persian rug.
   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.

   Links:
     * Mallet: Tribal Oriental Rugs

     * Dilucchio: Navajo 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.

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).

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 centre, 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.

   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:"
   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.

   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.

Symmetry in other arts and crafts

   Celtic knotwork
   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.

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.

   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 an 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.

   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).

   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, colour, 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.

   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.

Symmetry in games and puzzles

     * See also symmetric games.

     * See sudoku.

   Puzzles
     * The Diamond 16 Puzzle

   Board games
     * The Symmetrical Chess Collection

Symmetry in literature

   See palindrome.

Moral symmetry

     * Tit for tat
     * Reciprocity
     * Golden Rule
     * Empathy & Sympathy
     * Reflective equilibrium

   Retrieved from " http://en.wikipedia.org/wiki/Symmetry"
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