# 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. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations.

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.

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Again symmetry group D4, created from the previous image by shifting each quadrant to the opposite corner

## Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × XX, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: VV acting on the set of functions x: VW by (gx)(v)=x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v)=x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v)=w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself.

In 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.

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.

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.

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 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.

### 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:

## Translational symmetry

See main article translational symmetry.

Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(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 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.

## Screw axis symmetry

In 3D, screw axis symmetry is invariance under a rotation about an axis combined with translation along that axis .

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)

## 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 or the diamond theorem.

Compare the modified symmetry model for vector fields, above.

## 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 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.

## 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.

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.

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

## 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.

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

## 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" (∧, ${\displaystyle \land }$, or &), "or" (∨), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").

## 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.

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

## Symmetry in physics

(see main article: symmetry in physics)

Symmetry in physics has been generalized to mean invariance under any kind of transformation. This 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); and also, Wigner's theorem, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

## 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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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.

### 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.

### 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.

### 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.

### 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). 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 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.

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 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

File:Celticknotwork.png
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).

### Aesthetics

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.

Puzzles

Board Games

See palindrome.

## 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.

## 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.
• Weyl, Hermann (1952). Symmetry. Princeton University Press. ISBN 0-691-02374-3.