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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. Whether there is symmetry may depend on the properties under consideration: for a 2D image we may consider just the shapes, or also the colors; for an object, we may also consider density, chemical composition, etc.
These operations which do not appear to change an object form a group, the symmetry group of the object.
Again symmetry group D 4, created from the previous image by shifting each quadrant to the opposite corner
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.
Given some symmetry, the properties of part of the object, the fundamental domain, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, we need one point in every equivalence class to define the full object.
Any figure (not necessarily confined to the fundamental domain) can be converted to one with a desired symmetry by "adding" copies which are transformed by the symmetry operation(s). For this purpose it does not matter what the result is of adding e.g. red, red, and blue, as long as the addition is commutative and associative.
Mirror-image symmetry
A reflection "flips" an object over a line (in 2D) or plane (in 3D), inverting it to its mirror image, as if in a mirror. If the result is the same then we have mirror-image symmetry (also known in the terminology of modern physics as P-symmetry).
This is the most familiar and conventionally taught type of symmetry. It applies for instance for the letter T: when this letter 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."
For each line or plane of reflection, the symmetry group is isomorphic with C2. The fundamental domain is a half-plane or half-space.
See also axis of symmetry, plane of symmetry.
Rotational symmetry
A rotation rotates an object about a point (in 3D: about an axis). Rotational symmetry of order n, also called n-fold rotational symmetry, with respect to a particular point or axis means that rotation by an angle of 360°/n (180°, 120°, 90°, 72°, 60°, 51 3/7 °, etc.) does not change the object. (If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.) For each point or axis of symmetry the symmetry group is isometric with the cyclic group Cn of order n. The fundamental domain is a sector of 360°/n.
Examples without additional mirror image symmetry:
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The properties of the object on a radial half-line fully define such an object.
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. The properties of the object on a half-plane through the axis, and on a radial half-line, respectively, fully define such objects. An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).
Translational symmetry
A translation "slides" an object from one area to another by a vector. Translational symmetry of an object implies that, at least in one direction, it is infinite: for any given point, the set of points with the same properties due to the translational symmetry form an unbounded lattice. The properties of a line segment (1D), an infinite strip (2D) or a slab (3D), such that the vector starting at one side ends at the other side, fully define such objects. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector.
In 2D there may be double translational symmetry, given by two vectors. This implies that the object is infinite in both dimensions. The representation is not unique, instead of a and b we can also take a and a-b, etc. In general, we can take pa + qb and ra + sb for integers p, q, r, and s such that ps-qr is 1 or -1. This ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair.
Each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental domain: any pattern on it is possible, and this fully defines the whole object.
Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side.
For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group p1 (the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have p2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one.
Similarly, in 3D there may be double or triple translational symmetry.
In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane (cross-section) or line, respectively, fully defines the whole object.
For each set of k independent translation vectors the symmetry group is isomorphic with Zk
An example of translational symmetry (frieze group nr. 1) is:
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(get the same by moving one line down and two positions to the right), and of two-fold translational symmetry (wallpaper group p1):
* |* |* |* |
|* |* |* |*
|* |* |* |*
* |* |* |* |
|* |* |* |*
|* |* |* |*
(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 mirror-image symmetry nor rotational symmetry.
Compare periodic function.
Glide reflection symmetry
With glide reflection symmetry a combination of a reflection and a translation results in the same image (not to be confused with cases where each gives the same image). It implies translational symmetry, because applying a glide reflection twice corresponds to just a translation over twice the distance.
For each combination of a line of reflection and a translation vector the symmetry group is isomorphic with Z. Example:
* * * * * * * * * * * * * *
* * * * * * * * * * * * * *
Symmetry combinations
More complex symmetries are combinations of reflectional, rotational, translational, and glide reflection symmetry.
Mirror-image symmetry in combination with n-fold rotational symmetry, with the point of symmetry on the line of symmetry, implies mirror-image symmetry with respect to lines of reflection rotated by multiples of 180°/n, i.e. n reflection lines which are radially spaced evenly (for odd n this already follows from applying the rotational symmetry to a single reflection axis, but it also holds for even n). The symmetry group is the dihedral group of order 2n. For n > 2 an example is the n-sided regular polygon and various n-sided star polygons, including complex ones, which are a combination of simple ones for a divisor of n; also we have the simple "star" of n radial line segments (for even n this is a degenerate star polygon, for odd n it is not). Also multiple regular n-sided polygons with common center, differing by arbitrary rotations, as long as these rotation angles have mirror-image symmetry, e.g. two squares differing by a rotation angle of 10°, or three squares differing by two successive rotation angles of 10°.
Other examples:
Conversely, mirror-image symmetry with respect to two lines of reflection at an angle of 180°/n implies n-fold rotational symmetry (kaleidoscope effect).
In particular:
- Mirror-image symmetry with respect to two perpendicular lines of reflection implies rotational symmetry at the point of intersection for an angle of 180°. The symmetry group is the Klein four-group. This applies e.g. for a rectangle, a rhombus, and the letter H.
- Mirror-image symmetry of a square (with a pattern) with respect to the horizontal axis and to one diagonal, implies mirror-image symmetry with respect to the vertical axis and the other diagonal, and 4-fold rotational symmetry.
Mirror-image symmetry in combination with 2-fold rotational symmetry, with the point of symmetry not on the line of symmetry, implies an infinite sequence of alternating centers of symmetry and parallel lines of reflection, evenly spaced, with all these centers on a line perpendicular to the lines of reflection (the lines of reflection are the perpendicular bisectors of the line segments between adjacent copies of the points of symmetry). It also implies translational symmetry with as translation vector twice the difference in position between adjacent centers. This is frieze group nr. 6.
Translational symmetry can only be combined with 2-, 3-, 4-, and 6-fold rotational symmetry (angles of 180°, 120°, 90°, and 60°), see crystallographic restriction theorem. In these cases the translational symmetry applies along lines in 1, 3, 2, and 3 directions, respectively. This applies for 13 of the 17 wallpaper groups.
In the case of translational symmetry combined with 2-fold rotational symmetry, other centers of this symmetry can be found by translations by half the distances (the linear or 2D grid of rotocenters is twice as dense in each dimension as that of replicas of any given point by translation).
n-fold rotational symmetry with respect to two points of rotation implies translational symmetry.
Mirror-image symmetry in combination with translational symmetry
Mirror-image symmetry in combination with translational symmetry, with the translational vector not along the line of reflection, implies that there are infinitely many parallel lines of reflection, with a spacing such that one half of the translational vector, starting at one, ends at the next.
With translation in one direction, this is freeze group 4, or in the case of additional symmetry, 6 or 7.
With translation in two directions there are two cases:
- the translation vectors can be chosen to be perpendicular, and the rectangle spanned by these can be positioned with the axes of reflection along two opposite sides and halfway
wallpaper group cm (also called *x); the rhombus inside is the translation cell
- the second case concerns wallpaper group cm (and in the case of additional symmetry: cmm); one can choose different representations:
- the translation vectors can be chosen symmetrically with respect to an axis of reflection; then they have equal magnitude and the rhombus spanned by these has axes of reflection along a diagonal and through the other two vertices
- we can also choose one translation vector perpendicular to the axes of reflection, while they cross it at the ends and midway; then the other translation vector can be chosen such that it ends at the axis of reflection crossing the first translation vector midway; in that case the two span a parallelogram with one diagonal having equal length as each of one pair of sides (hence it is composed of two isosceles triangles) with the axes of reflection through all vertices
-
one can consider a rectangle with one pair of sides perpendicular to the axes of reflection (while, again, they cross it at the ends and midway) and the other pair of sides parallel to it; in that case the latter are not translation vectors; from a vertex to halfway a side of the first pair is the other translation vector; such a rectangle (in the figure the left and right half of the full rectangle), reproduced by translation, fills the plane and forms a common tiling (see the picture of the brick wall; in relation to the upper image and the description, the image is rotated 90°, and the bricks in the image are horizontally symmetric); due to the symmetry, one half of it is a fundamental domain; this can e.g. be rectangular (one quadrant of the full image at the top, one half of a brick). If the bricks are vertically symmetric the brick's image without rotation represents another correspondence with the upper image, with the brick in the two strips in the center.
wallpaper group cmm (also called 2*22); the rhombus of dark blue markers is the translation cell; note that the lattice of rotational centers is twice as dense in both directions as that of the translations (see also below)
.
Group cm can also be described as a rectangular checkerboard pattern, where the pattern of each of the two tiles is symmetric in, say, the horizontal direction, or looking at it differently (by shifting half a tile) a checkerboard pattern where the two tiles are each other's mirror image.
With additional reflection axes perpendicular to the other ones, we have cmm; in the case of the bricks this corresponds to homogeneous bricks, or, more generally, double symmetric ones.
Group cmm can be described as a checkerboard pattern of 2-fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile in both directions) a checkerboard pattern of two horizontally and vertically symmetric tiles.
Rotational symmetry of order 3 and/or 6 in combination with translational symmetry
Of course rotational symmetry of order 3 or more in combination with translational symmetry implies translational symmetry in two directions.
Rotational symmetry of order 3 at one center of rotation and order 2 at another one implies rotational symmetry of order 6 at some center of rotation.
p6m image showing (also for p6) the centers of rotation of order 6 (centers of the hexagons), the centers of rotation of order 3 (centers of the triangles), and in particular the positions of the centers of rotation of order 2 (vertices)
In the case of rotational symmetry of order 6, the centers of rotation of order 3 are arranged in a honeycomb structure, and the centers of rotation of order 2 in little triangles around them, touching each other, and also forming hexagons, rotated 30° and a little smaller.
Rotational symmetry of order 4 in combination with translational symmetry
tilted and shifted version of p4g, showing a larger square
Of course rotational symmetry of order 4 in combination with translational symmetry implies translational symmetry in two directions.
There are two different rotational centers of order 4, each in an upright square lattice, and together in a denser diagonal square lattice (orientations are expressed relative to the translational cells), each as many as there are translational cells. Also there is one kind of rotational center of order 2, there are as many of them as the other two together.
In the figures the two kinds of rotational centers of order 4 are distinguished by color (red and green), except in p4g, where the two kinds are each other's mirror image, both shown in green.
There is, of course, also translational symmetry with translations √2 times as large as the minimum, diagonally. Therefore the symmetries mentioned in the previous paragraph also apply in these larger translational squares. The two rotational centers of order 4 mentioned there are of the same kind in the larger squares, and the rotational centers midway on the sides are also of order 4.
Only in group p4g (4*2) the properties really change when considering these larger, tilted squares: the lines of reflection, which were in diagonal direction, are horizontal and vertical relative to the larger squares, positioned at 1/4 and 3/4 of the square. The rotational centers midway on the sides of the larger squares are the mirror image of those in the corners and in the center. (The last image shows a version which is shifted 1/4 of the large square.)
In p4g there is a checkerboard pattern of 4-fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile) a checkerboard pattern of horizontally and vertically symmetric tiles and their 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).
Color
When colors are ignored there may be 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.
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.
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.
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.
In fact, prior to the 20th century, groups were synonymous with transformation groups (i.e. group actions). It's only during the early 20th century that the current abstract definition of a group without any reference to group actions was used instead.
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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; including reduction by symmetry of the energy functional under the action by a group and 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, facial 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.
 
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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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.
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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.
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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.
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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 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 |
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D# |
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E |
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F |
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F# |
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G |
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G# |
| D |
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C# |
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C |
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B |
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A# |
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A |
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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 |
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3 |
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4 |
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5 |
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6 |
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7 |
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8 |
| 2 |
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1 |
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0 |
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11 |
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10 |
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9 |
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8 |
| 4 |
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4 |
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4 |
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4 |
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4 |
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4 |
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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.
  
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).
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 literature
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.
Moral symmetry
Related topics
External links
References
- 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
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