# Symmetry

**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. A rotation rotates an object using a point as its center. A translation "slides" an object from one area to another by a vector. Even more complex operations on a geometric object, like shrinking or shape warping, can be reduced to the operation of translation of every point within the object. 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 might be responsible for the mild altered state of consciousness one gets by observing intricate patterns based on symmetry.

## Symmetry in geometry

The most familiar and conventionally taught type of symmetry is the left-right 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 but no reflectional symmetry is the swastika.

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, coined by Mandelbrot is symmetry involving scale. 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. Tiling*

## Symmetry in mathematics

An example of a mathematical expression exhibiting symmetry is *a*^{2}*c* + 3*ab* + *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 well-hidden symmetries of fields. See also symmetric function.

## 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.

## 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 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 and cosmology.

## Symmetry in biology

*to be written*

## Symmetry in the arts and crafts

You can find the use of symmetry across a wide variety of arts and crafts.

### Architecture

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.

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### 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 cross-hatchings.

Links:

- Chinavoc: The Art of Chinese Bronzes
- Grant: Iranian Pottery in the Oriental Institute
- The Metropolitan Museum of Art - Islamic Art

### Quilts

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.

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### Carpets, rugs

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.

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### 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 scaless 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-diminised 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 centerss.Perle (1992) explains "C-E, D-F#, 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-2 family (with G# equal to 0).

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 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)

#### Equivalencey

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically.See also: Asymmetric rhythm

#### Sources

- Perle, George (1990).
*The Listening Composer*, p. 112. California: University of California Press. ISBN 0520069919. - Perle, George (1992). Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 81-96

#### External links

### 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.

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See also M. C. Escher, wallpaper group, tiling.### Literature

- Skaalid: Design Theory
- Williams: Symmetry, Design and Patterns
- Mathforum: Symmetry/Tesselations
- Calotta: A World of Symmetry
- Dutch: Symmetry Around a Point in the Plane
- Sanders: Transformations and Symmetry

## 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 relationship between the bandwidth allocated for data sent and received through that product.

The majority of internet services used by residential customers would be *asymmetrical*. This is due to the fact that when requesting content from a server the amount of data sent to the server is normally far less that the amount of information returned by the server.