Encyclopedia
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 of the other half.
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 .
Symmetries may also be found in living organisms including humans and other animals .
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
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
=x . 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=x 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=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
=h where
h rotates any vectors and pseudovectors in
x, and inverts any vectors according to rotation and inversion in
g, see symmetry in physics. The symmetry group of
x consists of all
g for which
x=h 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, 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+, the group of direct isometries. 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 , 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
- take for each orbit some kind of average or sum of the values of x at the points of the orbit
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 .
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 is, in the direction of the cylinder, constant, but nonzero. For vectors 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 acts transitively on all these points, while 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 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.
Rotational symmetry
See
rotational symmetry.
Translational symmetry
See main article
translational symmetry.
Translational symmetry leaves an object invariant under a discrete or continuous group of
translationsTa =
p +
aGlide reflection symmetry
A
glide reflection 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 with symmetry group S2n of order 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.
- Cnh ; 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.
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
See also space group.
Symmetry combinations
See symmetry combinations.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 . The new image may have more symmetry. Also colors may provide a special kind of symmetry, e.g. with corresponding points having opposite colors , such as in the
yin and yang symbol or the .
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 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.
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 . This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry . 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
- it has a mirror image counterpart
Symmetry in mathematics
- '
An example of a mathematical expression exhibiting symmetry is
a2c + 3
ab +
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
See also symmetric function, duality .
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" , "
or" , "biconditional" ,
NAND ,
XOR , and
NOR .
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
'
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, as it has become evident that practically all laws of nature originate in symmetries. See Noether's theorem ; 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 biology
See
symmetry and facial symmetry.
Symmetry in chemistry
See
Spectroscopy,
Molecular orbitalSymmetry in the arts and crafts
You can find the use of symmetry across a wide variety of arts and crafts.
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:
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:
As quilts are made from square blocks with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.
Links:
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:
Form
Symmetry has been used as a formal constraint by many composers, such as the arch form used by
Steve Reich,
Béla Bartók, and James Tenney . In classical music, Bach used the symmetry concepts of permutation and invariance; see .
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 , 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 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 .
| + | 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 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 Varčse, 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 .
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 .
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.
Symmetry in games and puzzles
- See also symmetric games.
- See sudoku.
Puzzles
Board Games
Symmetry in literature
See palindrome.
Symmetry in telecommunications
Some telecommunications services 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
- Tit for tat
- Reciprocity
- Golden Rule
- Empathy & Sympathy
- Reflective equilibrium
See also
References
- Livio, Mario . 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 . The Listening Composer, p. 112. California: University of California Press. ISBN 0-520-06991-9.
- Perle, George . Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 , pp. 81-96.
- Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
- Weyl, Hermann . Symmetry. Princeton University Press. ISBN 0-691-02374-3.
- Hahn, Werner . World Scientific. ISBN 981-02-2363-3.
- , published by Symmetrion, Budapest. ISSN 0865-4824.
External links
