Symmetry
From Academic Kids

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. The three main symmetrical operations are reflection, rotation, and translation. A reflection "flips" an object over a line, inverting it to its mirror image, as if in a mirror.
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A rotation rotates an object using a point as its center. An equilateral triangle has rotational symmetry with respect to an angle of 120°. Pentamerism is a body symmetry exhibited primarily by starfish; it is rotational symmetry with respect to an angle of 72°.
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates.
A translation "slides" an object from one area to another by a vector. More complex symmetries may be built by combining these operations. A particularly colorful example can be seen in the regular, repeated patterns of the wallpaper group. Symmetry occurs in geometry, mathematics, physics, biology, art, literature (palindromes), etc.
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 unrecognizable 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.
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Symmetry in geometry
The object with the most symmetry is empty space because any part of can be rotated, reflected or translated without apparent change.
The most familiar and conventionally taught type of symmetry is the leftright or mirror image symmetry exhibited for instance by the letter T: when this letter is reflected along a vertical axis, it appears the same.
An equilateral triangle exhibits such a reflection symmetry along three axes, and in addition it shows rotational symmetry: if rotated by 120 or 240 degrees, it remains unchanged. An instance of a shape which exhibits only rotational symmetry (with respect to an angle of 90 degrees) but no reflectional symmetry is the swastika.
An example of translational symmetry is:
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(get the same by moving one line down and two positions to the right), and of twofold translational symmetry:
* * * *  * * * * * * * * * * * *  * * * * * * * *
(get the same by moving three positions to the right, or one line down and two positions to the right; consequently get also the same moving three lines down).
In both cases there is neither mirrorimage symmetry nor rotational symmetry.
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.
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.
 Note: this needs a short paragraph about symmetrical patterns that completely cover a surface; e.g. the wallpaper group and tilings, but also aperiodic but nontheless symmetric tilings such as the Penrose tiling. Note that the Penrose tiling is a projection of a three dimensional lattice (which has cubic symmetry) down to two dimensions; thus, the readily appearent symmetry in three dimensions is rather hidden and obfuscated when seen in two. And this is, perhaps, an example of the grand lesson of finding symmetry in nature.
Symmetry in mathematics
An example of a mathematical expression exhibiting symmetry is a^{2}c + 3ab + b^{2}c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication.
In mathematics, one studies the symmetry of a given object by collecting all the operations that leave the object unchanged. These operations form a group. For a geometrical object, this is known as its symmetry group; for an algebraic object, one uses the term automorphism group. The whole subject of Galois theory deals with wellhidden symmetries of fields. See also symmetric function.
In fact, prior to the 20^{th} century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20^{th} century that the current abstract definition of a group without any reference to group actions was used instead.
Links:
 Perfect Symmetry Number Theory (http://www.symmetryperfect.com)
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.
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.
Also, physicists have come up with other generalizations like supersymmetry and quantum groups.
Symmetry in physics
The generalisation of symmetry in physics to mean invariance under any kind of transformation has become one of the most powerful tools of theoretical physics. See Noether's theorem(which, as a gross oversimplification, states that for every symmetry law, there is a conservation law) for more details. This has led to group theory being one of the areas of mathematics most studied by physicists; spontaneous symmetry breaking of transformations of symmetric groups appears to explain topics in particle physics (for example, the unification of electromagnetism and the weak force) and cosmology.
Symmetry in biology
See Bilateral symmetry, Pentamerism.
Symmetry in chemistry
See Group Theory, Spectroscopy, Molecular orbital
Symmetry in the arts and crafts
You can find the use of symmetry across a wide variety of arts and crafts.
Architecture
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Leaning Tower of Pisa
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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.
Links:
 Williams: Symmetry in Architecture (http://members.tripod.com/vismath/kim/)
 Aslaksen: Mathematics in Art and Architecture (http://www.math.nus.edu.sg/aslaksen/teaching/mathartarch.shtml)
Pottery
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 crosshatchings.
Links:
 Chinavoc: The Art of Chinese Bronzes (http://www.chinavoc.com/arts/handicraft/bronze.htm)
 Grant: Iranian Pottery in the Oriental Institute (http://wwwoi.uchicago.edu/OI/MUS/VOL/NN_SUM94/NN_Sum94.html)
 The Metropolitan Museum of Art  Islamic Art (http://www.metmuseum.org/collections/department.asp?dep=14)
Quilts
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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 (http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm)
Carpets, rugs
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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:
 Mallet: Tribal Oriental Rugs (http://www.marlamallett.com/default.htm)
 Dilucchio: Navajo Rugs (http://navajocentral.org/rugs.htm)
Music
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).
Pitch structures
Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of nonsymmetrical 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 (diminisheddiminished 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 nontonal tonal centers.
Perle (1992) explains "CE, DF#, [and] EbG, are different instances of the same interval...the other kind of identity...has to do with axes of symmetry. CE 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 interval4 family, CE is also a part of the sum4 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 nondiatonic. 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 Varese, 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.
Other arts and crafts
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).
Aesthetics
Symmetry does not by itself confer beauty to an object  many symmetrical designs are boring or overly challenging. 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.
Symmetry in literature
 to be written
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.
Moral symmetry
Related topics
 Symmetry groups
 Chirality
 Bilateral symmetry
 Radial symmetry
 Pentamerism
 Spontaneous symmetry breaking
 Gödel, Escher, Bach
 M. C. Escher
 Wallpaper group
 Tiling
 Asymmetric rhythm
External links
 (A)symmetrical wallpaper tiles (http://0waldo.com) by Walter Muncaster (user:0waldo)
 An Analysis of the first movement of the Fourth String Quartet (1928) (http://home.earthlink.net/~akuster/music/bartok/quartet4.htm) by Andrew Kuster
 Skaalid: Design Theory (http://www.usask.ca/education/coursework/skaalid/theory/theory.htm)
 Williams: Symmetry, Design and Patterns (http://www.uh.edu/hti/curriculum_units/1999/v05/07.htm)
 Mathforum: Symmetry/Tesselations (http://mathforum.org/library/topics/sym_tess/)
 Calotta: A World of Symmetry (http://www.teachersnetwork.org/teachnet/westchester/symmetry.htm)
 Dutch: Symmetry Around a Point in the Plane (http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM)
 Sanders: Transformations and Symmetry (http://www.punahou.edu/acad/sanders/MathArt/MACch2sym.html)
 Saw: Design Notes (http://daphne.palomar.edu/design/conclude.html)
 Chapman: Aesthetics of Symmetry (http://home.earthlink.net/~jdc24/symmetry.htm)
 Abas: The Wonder Of Symmetry (http://www.bangor.ac.uk/~mas009/psym.htm)
 ISIS Symmetry (http://www.mi.sanu.ac.yu/~jablans/isis0.htm)
References
 Perle, George (1990). The Listening Composer, p. 112. California: University of California Press. ISBN 0520069919.
 Perle, George (1992). Symmetry, the TwelveTone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 8196de:Symmetrie
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